Step-Index and Graded-Index Optical Fibers

An optical fiber is where total internal reflection stops being a laboratory curiosity and becomes a wire made of light. Two results this chapter has already built are what a fiber runs on: aq-optics-total-internal-reflection-guiding shows that a ray trapped in a high-index slab bounces forever once its wall angle exceeds the critical angle $\theta_c=\arcsin(n_2/n_1)$ , and aq-optics-numerical-aperture-acceptance-angle turns that trapping into a cone — only light launched within a half-angle set by $\mathrm{NA}=\sqrt{n_1^2-n_2^2}$ is captured. A fiber wraps that slab into a cylinder: a core of index $n_1$ inside a cladding of slightly lower index $n_2$ , and light poured into the acceptance cone runs down the axis for kilometres. Everything interesting comes from one design choice — how the index falls from core to cladding. Leave it a cliff and you have a step-index fiber whose rays zig-zag off a hard wall. Let it slope down smoothly, ideally as a near-parabola, and you have a graded-index fiber whose rays never touch a wall: the falling index refracts them back toward the axis so they glide along sinusoids, as if the glass were a lens smeared along its length. That difference decides the property that matters most for carrying information — modal dispersion, the spread in arrival times among the paths light can take. Push the core small enough and only one path survives; that single-mode limit, set by $V<2.405$ , is the fiber that circles the planet.

Key takeaways

A fiber guides light in a high-index core ( $n_1$ ) ringed by lower-index cladding ( $n_2$ ) — total internal reflection wrapped into a cylinder — and the acceptance cone is fixed by $\mathrm{NA}=\sqrt{n_1^2-n_2^2}=n_1\sqrt{2\Delta}$ .
A step-index profile is constant $n_1$ then an abrupt drop to $n_2$ ; a graded-index profile lets $n(r)$ fall smoothly, ideally as a parabola, so rays are refracted back rather than reflected.
One power law spans the family: $n^2(r)=n_1^2[1-2\Delta(r/a)^{\alpha}]$ — $\alpha\to\infty$ step, $\alpha=2$ parabolic.
Graded-index wins on bandwidth because off-axis rays travel farther but through faster, lower-index glass; the optimal profile cancels the two, shrinking modal delay from $\propto\Delta$ to $\propto\Delta^2$ — a thousand-fold gain.
The normalized frequency $V=(2\pi a/\lambda_0)\,\mathrm{NA}$ counts modes ( $M\approx V^2/2$ step); $V<2.405$ forces single-mode operation and kills modal dispersion.
Key idea: both fiber types keep light where the index is highest — step-index with a hard reflecting wall (TIR), graded-index with continuous refraction — and that difference is exactly why graded-index fibers nearly cancel modal dispersion.

The core insight

Both fiber types guide by keeping light where the index is highest; step-index does it with a hard reflecting wall (TIR), graded-index does it with continuous refraction — and that difference is exactly why graded-index fibers largely cancel modal dispersion.

Where this lives today

The two profiles divide the modern world of fiber between them. Long-haul and access networks run on step-index single-mode fiber — a core barely $8$ – $10\,\mu\mathrm{m}$ across with $\Delta\approx0.3\%$ , engineered so $V<2.405$ across the $1.31$ and $1.55\,\mu\mathrm{m}$ windows. With one mode there is no modal dispersion; the remaining enemy is chromatic dispersion, managed with dispersion-shifted profiles and coherent receivers. This is the fiber in transoceanic cables and the strand entering a home; wavelength-multiplexed, one such fiber carries tens of terabits per second. Inside datacenters the economics invert, and graded-index multimode fiber (OM3/OM4/OM5, $50\,\mu\mathrm{m}$ core) wins. Its large core couples cheaply to low-cost 850 nm VCSEL transceivers, and the near-parabolic $\Delta^2$ delay scaling keeps the modal spread small enough for tens of gigahertz-kilometres over the few-hundred-metre reaches a server hall needs. The same self-focusing physics reappears as the GRIN micro-lens — a quarter-pitch parabolic rod that collimates a laser or fiber tip with no polished surface — used by the millions in couplers, endoscopes, and switch collimators.

The physical picture

Picture a marble rolling down a long tube. A step-index fiber is a straight-walled pipe: the marble runs in straight lines and slams off the hard walls, and one launched steeply bounces more often and rolls a longer path than one skimming the axis. A graded-index fiber is a pipe whose floor curves up into a valley; the marble is steered smoothly back to the bottom and never strikes an edge — its track is a gentle wave. Light does exactly this: in the step fiber a ray reflects whenever it meets the wall above the critical angle, while in the graded fiber the dropping index bends it continuously, so every ray oscillates about the axis with the same spatial period regardless of launch angle.

Figure

Step-index versus graded-index guiding. Left: a meridional ray zig-zags off the hard core–cladding wall at the critical angle θc ≈ 82°, and only rays inside the acceptance half-angle θa ≈ 11.5° are captured. Right: the falling index refracts every ray back toward the axis, so paths are sinusoids with a common period

The reason graded-index fiber carries more information is a beautiful compensation. In a step-index fiber every ray moves at $c/n_1$ , so the steep bouncing ray, with its longer path, arrives late and a pulse smears out. In a graded-index fiber the steep ray wanders into the outer glass where the index is lower and the local speed $c/n(r)$ is higher; the extra distance is run at extra speed, and with the right profile all rays arrive together to first order. In wave language a fiber is a potential well for light: the transverse index profile binds only a discrete set of guided modes, just as a quantum well admits only certain bound states, and a shallow enough well (small $V$ ) keeps only the ground state — the single-mode regime. The whole engineering trade then lives in one small number, $\Delta\sim0.01$ .

Figure

The index profile family n(r) for a 50 µm core, n₁ = 1.490 on axis dropping to n₂ = 1.477 at the edge (Δ = 0.9 percent, NA = 0.20). Step α → ∞ is the red rectangle, parabolic α = 2 the blue curve, triangular α = 1 the amber line — the single power law n²(r) = n₁²(1 − 2Δ(r⁄a)^α) spans all three

The core equations

A single power law captures every profile at once — the exponent $\alpha$ selects the fiber type (sharp cliff or smooth valley) while $\Delta$ and the core radius $a$ carry the glass parameters. It is the equation the profile figure is drawn from.

n^2(r)=n_1^2\left[1-2\Delta\left(\frac{r}{a}\right)^{\alpha}\right],\quad r\le a \qquad (1)

where: $n(r)$ — index at radius $r$ ; $n_1$ — on-axis (maximum) core index; $a$ — core radius; $\alpha$ — profile exponent ( $\alpha\to\infty$ step, $\alpha=2$ parabolic); $\Delta$ — fractional index difference. The index peaks at $n_1$ on the axis and sags to $n_2=n_1\sqrt{1-2\Delta}$ at $r=a$ ; the shape of that sag is the whole story. The fractional index difference is the fiber's master parameter — a pure number, usually a fraction of a percent, tying light-gathering, mode count and dispersion together. Its exact and weak-guidance forms agree because core and cladding indices differ so little.

\Delta=\frac{n_1^2-n_2^2}{2n_1^2}\approx\frac{n_1-n_2}{n_1} \qquad (2)

where: $n_2$ — cladding index; $\Delta$ — relative index step, typically $0.002$ to $0.02$ . The two forms coincide because $n_1+n_2\approx2n_1$ under weak guidance; $\Delta$ is roughly the fractional height of the cliff. The numerical aperture translates the index step into an acceptance cone — the sine of the largest half-angle from which the fiber accepts light. It follows from Snell's law at the endface plus the critical-angle condition inside, derived below.

\mathrm{NA}=n_0\sin\theta_{a}=\sqrt{n_1^2-n_2^2}=n_1\sqrt{2\Delta} \qquad (3)

where: $\mathrm{NA}$ — numerical aperture; $n_0$ — launch-medium index (air, $\approx1$ ); $\theta_a$ — acceptance half-angle; the wall critical angle is $\theta_c=\arcsin(n_2/n_1)$ . A larger $\mathrm{NA}$ gathers more light — good for coupling cheap sources, bad for dispersion. The normalized frequency, or $V$ -number, is the one dimensionless combination that governs how many modes a fiber supports, bundling core size, wavelength and NA into a single knob.

V=\frac{2\pi a}{\lambda_0}\,\mathrm{NA}=\frac{2\pi a}{\lambda_0}\,n_1\sqrt{2\Delta} \qquad (4)

where: $V$ — normalized frequency; $\lambda_0$ — free-space wavelength; $a$ — core radius. $V$ measures the core in "wavelengths across the acceptance cone": a fat core, short wavelength or high NA push $V$ up and admit more modes. For large $V$ the mode count follows from counting how many transverse ray directions fit the guiding condition — a phase-space estimate derived below — and grading lowers the count by a factor set by the exponent.

M\approx\frac{V^2}{2}\ (\text{step}),\qquad M\approx\frac{\alpha}{\alpha+2}\,\frac{V^2}{2}\ (\text{graded}) \qquad (5)

where: $M$ — guided-mode count, both polarizations; $\alpha$ — profile exponent, so parabolic $\alpha=2$ gives $M\approx V^2/4$ , half the step count. Fewer modes means less arrival-time spread — mode count is what a designer trades against bandwidth. When $V$ falls below the first zero of $J_0$ , every higher mode is cut off and only the fundamental survives. That threshold defines single-mode operation and, read as a wavelength, the cutoff wavelength above which a fiber is single-mode.

V<2.405\ \Rightarrow\ \text{single-mode},\qquad \lambda_c=\frac{2\pi a\,\mathrm{NA}}{2.405} \qquad (6)

where: $2.405$ — first zero of $J_0$ , cutoff of the $\mathrm{LP}_{11}$ mode; $\lambda_c$ — cutoff wavelength, above which only $\mathrm{LP}_{01}$ survives. A fiber is used above its cutoff to guarantee single-mode purity. Finally, the property that separates the two fiber types in practice: the modal delay spread per unit length, the arrival-time difference between fastest and slowest guided path. Step-index is linear in $\Delta$ ; optimally graded is quadratic — and that exponent is the whole reason graded-index fiber exists.

\frac{\Delta\tau}{L}\Big|_\mathrm{step}\approx\frac{n_1\Delta}{c},\qquad \frac{\Delta\tau}{L}\Big|_\mathrm{graded}\approx\frac{n_1\Delta^2}{8c} \qquad (7)

where: $\Delta\tau$ — arrival-time spread between fastest and slowest mode; $L$ — length; $c$ — vacuum speed of light. Because $\Delta\ll1$ , replacing $\Delta$ by $\Delta^2/8$ shrinks the spread by $\sim8/\Delta\sim10^3$ — metres of reach become kilometres.

Derivation ANumerical aperture from Snell's law and total internal reflection

The goal is to earn the acceptance-cone formula $\mathrm{NA}=\sqrt{n_1^2-n_2^2}=n_1\sqrt{2\Delta}$ rather than quote it, by tracking a single marginal ray from the air, through refraction at the flat endface, to its first meeting with the core–cladding wall, and demanding it strike that wall at exactly the critical angle. The just-guided ray defines the edge of the acceptance cone, and chasing it through two interfaces fuses Snell's law and the critical angle into one measurable number.

Step 1. A ray arrives at half-angle $\theta_a$ and refracts into the core at $\theta_r$ , so n_0\sin\theta_a=n_1\sin\theta_r. Why: the endface is an ordinary refracting surface, and Snell's law fixes how steeply the ray travels inside.

Step 2. Inside, the ray meets the wall at the complement of $\theta_r$ , and for the marginal ray this equals the critical angle: \theta_\mathrm{wall}=90^\circ-\theta_r=\theta_c. Why: a ray refracted more steeply hits the wall below $\theta_c$ and leaks out, so the just-guided ray sets $\theta_\mathrm{wall}=\theta_c$ .

Step 3. Substitute $\sin\theta_r=\cos\theta_c$ into Step 1: n_0\sin\theta_a=n_1\cos\theta_c=n_1\sqrt{1-\sin^2\theta_c}. Why: writing it as $\cos\theta_c$ lets us bring in the critical-angle condition $\sin\theta_c=n_2/n_1$ .

Step 4. Insert $\sin\theta_c=n_2/n_1$ and simplify: \mathrm{NA}=n_0\sin\theta_a=n_1\sqrt{1-(n_2/n_1)^2}=\sqrt{n_1^2-n_2^2}=n_1\sqrt{2\Delta}. Why: $n_1$ cancels into the difference of squares, and equation (2) exposes the weak-guidance scaling $\mathrm{NA}\propto\sqrt{\Delta}$ .

Go deeper · Derivation B · 1 minSinusoidal ray paths in a parabolic graded-index fiber

Derivation BSinusoidal ray paths in a parabolic graded-index fiber

The goal is to prove that in a parabolic profile every meridional ray, whatever its launch slope, oscillates about the axis with one common spatial period — the self-focusing property that both equalizes modal delay and makes a short GRIN rod a lens. We start from the eikonal ray equation, reduce it to the paraxial regime where slopes are small and $n\approx n_1$ , insert the parabolic profile, and recognize a harmonic oscillator in the propagation coordinate whose solution is a sine wave with a slope-independent frequency.

Step 1. A ray in a graded medium obeys the eikonal ray equation \frac{\mathrm{d}}{\mathrm{d}s}\!\left(n\,\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s}\right)=\nabla n, with $s$ the arc length. Why: this is the geometric-optics statement that rays bend toward higher index — the continuous analogue of Snell's law.

Step 2. For paraxial rays $\mathrm{d}s\approx\mathrm{d}z$ , $n\approx n_1$ , and the transverse component becomes n_1\,\frac{\mathrm{d}^2 r}{\mathrm{d}z^2}=\frac{\partial n}{\partial r}. Why: near the axis the ray advances almost purely in $z$ , so its transverse acceleration comes only from the radial index gradient.

Step 3. To first order in $\Delta$ the $\alpha=2$ profile is $n(r)\approx n_1[1-\Delta\,r^2/a^2]$ , with gradient \frac{\partial n}{\partial r}=-\frac{2n_1\Delta}{a^2}\,r. Why: a parabolic index makes the radial restoring force grow linearly with displacement — the ingredient a harmonic oscillator needs.

Step 4. Divide by $n_1$ : \frac{\mathrm{d}^2 r}{\mathrm{d}z^2}+\frac{2\Delta}{a^2}\,r=0. Why: this is $\ddot r+\Omega^2 r=0$ with $\Omega=\sqrt{2\Delta}/a$ — the ray obeys Hooke's law along the fiber.

Step 5. The solution is r(z)=r_0\cos(\Omega z)+\frac{r_0'}{\Omega}\sin(\Omega z),\qquad \Omega=\frac{\sqrt{2\Delta}}{a}. Why: since $\Omega$ is independent of the launch slope $r_0'$ , all rays share the period $2\pi/\Omega$ and re-focus together — the origin of near-zero modal delay and of the GRIN lens.

Go deeper · Derivation C · 1 minCounting modes in a step-index fiber by transverse phase space

Derivation CCounting modes in a step-index fiber by transverse phase space

The goal is to recover the workhorse estimate $M\approx V^2/2$ without solving the full cylindrical eigenvalue problem, by counting how many transverse ray directions satisfy the guiding condition inside the core. The trick is to treat the transverse plane as a two-dimensional phase space of position and wavevector, use the density-of-states result that each mode occupies a cell of area $(2\pi)^2$ , and cap the wavevector at the largest value that keeps a mode bound in the cladding.

Step 1. A mode is guided only if its propagation constant exceeds $k_0 n_2$ , i.e. k_T^2=k_0^2 n_1^2-\beta^2\le k_0^2\,(n_1^2-n_2^2). Why: a mode with too large a transverse wavenumber travels too steeply, drops below the cladding cutoff, and radiates away.

Step 2. The ceiling on transverse wavenumber is therefore k_{T,\max}=k_0\sqrt{n_1^2-n_2^2}=k_0\,\mathrm{NA}=\frac{V}{a}, using $V=k_0 a\,\mathrm{NA}$ . Why: this ties the cutoff to the measurable $V$ -number, so the count comes out in $V$ alone.

Step 3. Count states as phase-space area over cell size — transverse area $\pi a^2$ times a wavevector disc $\pi k_{T,\max}^2$ : N=\frac{(\pi a^2)(\pi k_{T,\max}^2)}{(2\pi)^2}=\frac{V^2}{4}. Why: this is Weyl counting — the density of states of a 2D box, applied to the fiber cross-section.

Step 4. Each spatial mode carries two polarizations, so M=2N=\frac{V^2}{2}. Why: doubling for polarization gives the step-index result; repeating against the graded profile inserts the factor $\alpha/(\alpha+2)$ , halving the count for a parabola.

Limiting cases

The power-law profile $n^2(r)=n_1^2[1-2\Delta(r/a)^\alpha]$ contains every fiber here as a special value of the exponent $\alpha$ or the index step $\Delta$ .

Case	Result	Meaning
Step-index ( $\alpha\to\infty$ )	$n(r)=n_1$ for $r<a$ , then $n_2$	Abrupt cliff; guiding by TIR. Simplest to make, but multimode dispersion $\propto\Delta$ is large.
Parabolic / optimal ( $\alpha\approx2$ )	$n^2(r)=n_1^2[1-2\Delta(r/a)^2]$	Rays are sinusoids; the optimum $\alpha_\mathrm{opt}\approx2(1-\tfrac{6}{5}\Delta)$ equalizes delays to $O(\Delta^2)$ — the workhorse multimode profile.
Triangular ( $\alpha=1$ )	$n^2(r)=n_1^2[1-2\Delta\,r/a]$	A linear taper; used in some dispersion-tailored designs, far from modal-optimal.
Weak guidance ( $\Delta\ll1$ )	$\mathrm{NA}\approx n_1-n_2,\ \theta_c\to90^\circ$	Typical silica; vector modes collapse into the scalar $\mathrm{LP}_{lm}$ set.
Small core ( $V<2.405$ )	$M=1$ (only $\mathrm{LP}_{01}$ )	Single-mode: modal dispersion vanishes, leaving only chromatic effects — the long-haul regime.

Worked examples

1 · Designing a standard single-mode telecom fiber (the confidence builder)

A silica fiber has core index $n_1=1.4675$ , cladding index $n_2=1.4623$ , and core radius $a=4.0\,\mu\mathrm{m}$ . Find its fractional index difference, numerical aperture, and cutoff wavelength, and confirm it is single-mode at both 1310 nm and 1550 nm.

Step 1. Index difference: $\Delta=(n_1^2-n_2^2)/(2n_1^2)=0.01524/4.30711=\mathbf{3.54\times10^{-3}}$ , about a third of a percent — a gently guiding fiber, as long-haul silica always is.

Step 2. Numerical aperture: $\mathrm{NA}=\sqrt{n_1^2-n_2^2}=\sqrt{0.01524}=\mathbf{0.123}$ , acceptance half-angle $\theta_a=\arcsin(0.123)\approx7.1^\circ$ ; the check $n_1\sqrt{2\Delta}=0.123$ agrees.

Step 3. Cutoff wavelength, equation (6): $\lambda_c=2\pi a\,\mathrm{NA}/2.405=(6.2832)(4.0\,\mu\mathrm{m})(0.123)/2.405=\mathbf{1.29\,\mu m}$ ; above it only $\mathrm{LP}_{01}$ survives.

Step 4. Operating windows, equation (4): $V(1310\,\mathrm{nm})=3.101/1.310=2.37$ and $V(1550\,\mathrm{nm})=3.101/1.550=2.00$ , both below $2.405$ . So the fiber is single-mode at both telecom wavelengths, NA = 0.123, cutoff λc ≈ 1.29 µm — a real G.652 profile.

2 · Why step-index multimode fiber cannot go far

A multimode fiber has $n_1=1.490$ , $\mathrm{NA}=0.20$ , and a $50\,\mu\mathrm{m}$ core ( $a=25\,\mu\mathrm{m}$ ), operated at 850 nm. Find $\Delta$ , the $V$ -number and mode count, and compare the modal delay spread over 1 km for a step-index versus an optimally graded profile.

Step 1. From $\mathrm{NA}=n_1\sqrt{2\Delta}$ : $\Delta=(\mathrm{NA}/n_1)^2/2=(0.20/1.490)^2/2=\mathbf{9.0\times10^{-3}}$ , so $n_2=\sqrt{n_1^2-\mathrm{NA}^2}=1.477$ .

Step 2. Normalized frequency: $V=2\pi a\,\mathrm{NA}/\lambda_0=(6.2832)(25\,\mu\mathrm{m})(0.20)/0.850\,\mu\mathrm{m}=\mathbf{37.0}$ — deep multimode. Step count $M\approx V^2/2=\mathbf{683}$ ; a parabolic fiber of the same $V$ carries $M\approx V^2/4=\mathbf{342}$ , half as many.

Step 3. Step-index modal delay, equation (7): $\Delta\tau/L=n_1\Delta/c=(1.490)(9.0\times10^{-3})/(3.0\times10^8)=4.47\times10^{-11}\,\mathrm{s/m}$ , i.e. ≈ 45 ns over 1 km — catastrophic for gigabit signalling.

Step 4. Optimally graded: $\Delta\tau/L=n_1\Delta^2/(8c)=(1.490)(9.0\times10^{-3})^2/(2.4\times10^9)=5.0\times10^{-14}\,\mathrm{s/m}$ , i.e. ≈ 50 ps over 1 km — a factor $8/\Delta\approx890$ smaller. Grading the same glass turns 45 ns/km into 50 ps/km, which is why datacenter links are graded-index.

3 · A graded-index rod as a lens

A parabolic GRIN rod has $\Delta=0.010$ and radius $a=25\,\mu\mathrm{m}$ . Find the ray period and the length that collimates a point source.

Step 1. From the ray solution, $\Omega=\sqrt{2\Delta}/a=\sqrt{0.020}/(25\times10^{-6})=5.66\times10^{3}\,\mathrm{m^{-1}}$ , so the period is $\Lambda=2\pi/\Omega=\mathbf{1.11\,mm}$ ; every ray re-crosses the axis after this distance.

Step 2. A point on the input face collimates after one quarter period, where diverging rays are just bent parallel: $L_{1/4}=\Lambda/4=\mathbf{0.28\,mm}$ . A 0.28 mm slice of this rod is a lens with no curved surfaces — the principle of the GRIN micro-lenses that pigtail lasers.

Physics of the portrait

In fiber optics the decisive advance was not a discovery but an act of stubborn accounting — proving the losses blamed on glass were removable impurities.

Origins · 1 minCharles K. Kao and how this was discovered

Charles K. Kao

1933–2018 · low-loss optical fiber for communication

In the mid-1960s optical fiber was a laboratory toy: glass fibers of the day lost about 1000 decibels per kilometre, so a signal died within metres, and the consensus was that glass was too lossy to carry light any distance. Charles Kuen Kao, at Standard Telecommunications Laboratories in England, refused it. With George Hockham he published in 1966 a meticulous analysis arguing that the loss was not intrinsic to silica but dominated by trace impurities — transition-metal ions and water — and that purified glass could fall below 20 dB/km, the threshold at which fiber would beat copper. He travelled to glassmakers to measure and cajole, insisting the target was reachable. It was: in 1970 Corning's Maurer, Keck and Schultz drew silica fiber at 17 dB/km, and losses soon reached the Rayleigh floor near 0.2 dB/km at $1.55\,\mu\mathrm{m}$ . Every idea here — the small single-mode core, the graded profile that equalizes modal delay, the $V$ -number that sets the mode count — became worth engineering only because Kao showed the light would arrive. He won the 2009 Nobel Prize in Physics, and today's undersea cables descend from his 1966 paper.

History

1842

— Daniel Colladon guides light along a falling jet of water by total internal reflection

1930

— Heinrich Lamm transmits a crude image through a bundle of unclad glass fibers

1954

— van Heel and Hopkins-Kapany introduce the clad fiber that suppresses cross-talk; Kapany soon coins the term fiber optics

1966

— Kao and Hockham argue that purified glass below 20 dB/km can carry telecommunications

1970

— Corning's Maurer, Keck and Schultz draw the first low-loss silica fiber at 17 dB/km

1973

— Gloge and Marcatili derive the optimal near-parabolic profile that equalizes modal delay

Going further

Texts. Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 9 (Fiber Optics) — primary source for the profile, $V$ -number, mode-count and GRIN-ray results · Hecht, Optics, 5th ed., §5.6 · G. P. Agrawal, Fiber-Optic Communication Systems, Ch. 2 · Demtröder, Laser Spectroscopy, Vol. 1, on optical waveguides · Loudon, The Quantum Theory of Light, on guided-mode quantization.

Provenance. The definitions of $\Delta$ , NA, the $V$ -number and the mode counts follow Saleh and Teich (Ch. 9) and Agrawal (Ch. 2). A verifier should recompute, and each checks out numerically here: (i) $\mathrm{NA}=\sqrt{n_1^2-n_2^2}=n_1\sqrt{2\Delta}$ from Snell plus TIR; (ii) the cutoff $V=2.405$ as the first zero of $J_0$ , whose eigenvalue origin is owned by aq-optics-waveguide-modes-wave-equation; (iii) the parabolic SHM equation $r''+(2\Delta/a^2)r=0$ and its sinusoidal solution; (iv) $M\approx V^2/2$ from phase-space counting, with the graded factor $\alpha/(\alpha+2)$ . Worked numbers use standard silica values ( $n\approx1.46$ – $1.49$ ) and the 850, 1310 and 1550 nm windows.

Prerequisites. Total Internal Reflection and Optical Guiding · Numerical Aperture and Acceptance Angle. Leads to. Waveguide Modes and the Wave Equation · Single-Mode and Multimode Operation · Chromatic and Modal Dispersion in Fibers · Fiber Attenuation and Loss Mechanisms.

Ask the tutor: Want to see how the $V$ -number and the $2.405$ cutoff fall out of solving the cylindrical wave equation, or how the optimal graded-index exponent is fine-tuned below $\alpha=2$ ?

Is the light really bouncing off a mirror inside a step-index fiber?

Not off a mirror, but off an index boundary. When a ray meets the core–cladding wall beyond the critical angle $\theta_c=\arcsin(n_2/n_1)$ , total internal reflection sends essentially all the light back into the core — no metallic surface, no interface loss. That is why fiber is so low-loss: confinement is a property of the two glasses, not of a coating that soaks up energy on every bounce. In a graded-index fiber there is no wall event at all — the falling index refracts the ray back before it ever reaches the cladding.

What is the V-number, in plain terms?

The $V$ -number, $V=(2\pi a/\lambda_0)\,\mathrm{NA}$ , measures the core in units of wavelength weighted by how strongly it guides — loosely, how many half-wavelengths of transverse structure fit across the core. A small $V$ means only one field pattern fits, so the fiber is single-mode; a large $V$ leaves room for many, and the count grows as $V^2/2$ . Bundling core radius, wavelength and NA into one number, $V$ predicts single- versus multimode behaviour without solving the wave equation each time.

If graded-index is better, why is long-haul fiber step-index single-mode?

Because grading solves a problem single-mode fiber does not have. Graded profiles exist to equalize the delays of many modes; a single-mode fiber carries only one, so there is no intermodal spread to cancel — its modal dispersion is zero. Forcing $V<2.405$ with a small core is a more complete cure than grading a multimode core, and easier to make as a step. Long-haul systems therefore use a small step-index core and fight only chromatic dispersion. Graded-index earns its keep only where a large, easily-coupled multimode core is wanted, as in short datacenter links.

Why is the single-mode cutoff exactly 2.405?

It is the first zero of the Bessel function $J_0$ , and it appears because the guided fields in a step-index core are Bessel functions of radius. Requiring the field to oscillate inside the core and decay in the cladding yields an eigenvalue equation whose lowest higher-order solution, the $\mathrm{LP}_{11}$ mode, first becomes guided when $V$ reaches this zero; below it, only $\mathrm{LP}_{01}$ remains. The full derivation belongs to the waveguide-modes article; here we take $2.405$ as the sharp threshold it provides for the step-index profile.

How can off-axis rays arrive at the same time if they travel farther?

This is the central trick of graded-index fiber, and it works because speed is not constant across the core. An off-axis ray follows a longer sinusoidal path, but that path swings out into the low-index glass near the edge, where $c/n(r)$ is higher. With a parabolic profile the extra distance and extra speed compensate almost exactly: to first order in $\Delta$ every ray's group delay matches, and the residual spread scales as $\Delta^2$ . Slow-but-short axial rays and long-but-fast off-axis rays arrive together — and getting the exponent slightly wrong degrades the cancellation, which is why it is engineered carefully.

Is single-mode fiber truly a single mode?

Almost — it carries one spatial mode, $\mathrm{LP}_{01}$ , but that mode has two orthogonal polarizations, so strictly it is two degenerate modes. In a perfectly circular, unstressed fiber they travel identically and the distinction is invisible. Real fibers are never perfectly round, so the two polarizations drift apart in speed — polarization-mode dispersion, a limiting impairment on long, fast links. So single-mode means one spatial pattern, not literally one propagating state; for everything here, treating it as the one $\mathrm{LP}_{01}$ pattern is exactly right.