Waveguide Modes and the Wave Equation

Total internal reflection tells you that light can be trapped inside a glass thread, but it cannot tell you what shape the trapped light takes, nor why a fibre carries only a handful of distinct beams instead of a continuum. That answer belongs to the wave equation. This article takes the source-free Helmholtz equation $\nabla^2\mathbf{E}+n^2k_0^2\mathbf{E}=0$ — the form Maxwell's equations collapse to inside a transparent dielectric — and asks one sharp question: which field patterns can travel down a guide of higher-index core and lower-index cladding without changing shape as they go? Demanding that the transverse profile translate rigidly along the axis, $\mathbf{E}\propto\mathbf{e}(x,y)\,e^{i(\beta z-\omega t)}$ , turns propagation into an eigenvalue problem. The eigenvalues are the propagation constants $\beta$ ; the eigenfunctions are the guided modes. What forces discreteness is boundary conditions. The field must be finite, continuous across each index step, and evanescent — decaying — in the cladding. Those requirements over-constrain the solution, so only special values of $\beta$ survive: a finite discrete set of guided modes atop a continuum of unguided radiation modes. In a planar slab the modes split cleanly into transverse-electric (TE) and transverse-magnetic (TM) families; in a circular fibre they are hybrid $\mathrm{HE}$ and $\mathrm{EH}$ modes that, when core and cladding indices are nearly equal, merge into the scalar linearly-polarized $\mathrm{LP}$ modes engineers actually use. One dimensionless number, the normalized frequency $V$ , counts them all, and its threshold value marks where a guide becomes single-mode — the foundation for dispersion, coupling, and every integrated-photonic device downstream.

Key takeaways

A waveguide mode is a field that propagates as $e^{i(\beta z-\omega t)}$ while keeping a fixed transverse profile — an eigenfunction of the transverse wave operator subject to the guide's boundary conditions.
Continuity plus an evanescent cladding tail quantizes the propagation constant into a discrete set; every guided mode obeys $n_2k_0<\beta<n_1k_0$ , travelling slower than a core plane wave but faster than a cladding one.
The eigenvalues follow from a transcendental equation — the slab TE relation $\tan(k_x a-m\pi/2)=\gamma/k_x$ , or the Bessel-matching relation in a fibre — and equivalently from a transverse round-trip resonance condition.
The normalized frequency $V=k_0a\,\mathrm{NA}$ counts modes: below $V=2.405$ (the first zero of $J_0$ ) a step-index fibre is single-mode; a symmetric slab is single-mode below $V=\pi/2$ and its fundamental never cuts off.
In the weak-guidance limit $n_1\approx n_2$ the vector hybrid modes degenerate into scalar $\mathrm{LP}_{lm}$ modes with a simple sinusoidal–Bessel profile — the practical basis for real fibres.
Key idea: a guided mode is an eigenfunction of the transverse wave operator — substitute $\mathbf{E}=\mathbf{e}(x,y)e^{i(\beta z-\omega t)}$ into the Helmholtz equation and demand bounded, continuous, evanescent-in-cladding solutions — so propagation becomes a boundary-value eigenvalue problem whose discrete eigenvalues $\beta$ are the only fields that keep their shape over long distances.

The core insight

A guided mode is an eigenfunction of the transverse wave operator: assume $\mathbf{E}(x,y,z,t)=\mathbf{e}(x,y)\,e^{i(\beta z-\omega t)}$ , substitute into the Helmholtz equation, and demand physically admissible (bounded, continuous, evanescent-in-cladding) solutions. This turns propagation into a boundary-value eigenvalue problem whose discrete eigenvalues $\beta$ are the only fields that keep their shape over long distances.

Where this lives today

The eigenvalue problem of this article is solved millions of times a day, mostly inside software. Every silicon-photonic circuit — the transceivers in a data centre, the modulators in a coherent link — is laid out with a numerical mode solver that discretizes exactly equation (2) over a waveguide cross-section and returns the effective indices and field profiles $\mathbf{e}(x,y)$ of each mode. Those $n_{\mathrm{eff}}$ values set how tightly a bend can turn before the mode radiates, how long a directional coupler must be for light to hop between two guides (coupled-mode theory is built directly on these eigenfunctions), and how a mode overlaps a fibre when the chip is packaged. High-index-contrast guides ( $n_1\approx3.5$ silicon in $n_2\approx1.45$ oxide) confine a mode to a cross-section well under a micron, which is what lets a photonic chip pack thousands of components. At the other end of the scale, the discreteness this theory predicts has become a resource rather than a limitation. Standard single-mode fibre such as Corning SMF-28 is engineered so its cutoff sits near $1.26\ \mu\mathrm{m}$ , guaranteeing one mode across the telecom bands; but the drive for capacity has revived multimode transmission in controlled form. Few-mode and multicore fibres deliberately guide a small handful of $\mathrm{LP}$ modes and treat each as an independent data channel — space-division multiplexing — recovering, mode by mode, the bandwidth that single-mode operation gave up. The very mode set Snitzer classified in 1961 is now the alphabet of a new generation of optical networks.

The physical picture

Start with the ray you already know. A ray zig-zags down the core, bouncing off the two interfaces by total internal reflection, and geometric optics would let it do so at any angle steeper than the critical angle. The wave picture forbids that. Treat the bouncing ray as a plane wave: as it crosses the core, reflects, and crosses back, it accumulates phase — the transverse propagation phase of two core crossings plus the phase jump of each total-internal-reflection bounce (the Goos–Hänchen shift). Unless that round-trip phase closes to a multiple of $2\pi$ , the wave interferes destructively with itself and washes out over a few bounces. Only a discrete set of ray angles survives, and the round-trip-resonance condition is, term for term, the same transcendental eigenvalue equation you get by matching fields at the boundaries. Ray and wave are one physics. Look at the same mode as a field. Across the core the transverse profile is a standing wave — a sinusoid with $m$ zeros for the $m$ -th mode — and outside the core it leaks a little way into the cladding as an exponentially decaying evanescent tail. Higher-order modes pack more oscillations across the core, so they carry a larger transverse wavevector $k_x$ and, since $k_x^2+\beta^2$ is fixed by the core index, a smaller axial $\beta$ ; that is why they travel at shallower angles and sit closer to cutoff. The bound $n_2k_0<\beta<n_1k_0$ is a "light-line" statement: $\beta$ must stay below the core light line (or the core field is evanescent, not oscillatory) and above the cladding light line (or the cladding field radiates instead of decaying). Guided modes live in the wedge between them.

Figure

The fundamental slab mode seen two ways. On the left a ray at incidence angle θ obeys total internal reflection and closes its transverse round-trip phase, giving β = n₁k₀ sin θ with core n₁ = 1.50 and cladding n₂ = 1.485. On the right the same mode is a cosine standing wave across the core that leaks into the cladding as evanescent tails decaying like e^(−γx), with effective index n_eff = β/k₀ = 1.492 lying between the two indices

Weak guidance — real telecom fibre, where $n_1$ and $n_2$ differ by under a percent — is what makes the whole thing tractable: the vector problem becomes scalar, the TE/TM/hybrid distinctions blur, and the modes collapse into the near-degenerate $\mathrm{LP}$ set. The clean way to see what a guide carries is to normalize: plot each mode's confinement $b=(n_{\mathrm{eff}}^2-n_2^2)/(n_1^2-n_2^2)$ against the single control parameter $V$ . Each mode appears as a curve that switches on at its own cutoff and climbs from loosely bound ( $b\to0$ ) to tightly bound ( $b\to1$ ), and the whole design of a fibre reduces to reading off which curves exist at the operating $V$ .

Figure

Normalized b–V diagram of a step-index fibre. Each guided LP mode is a curve of normalized index b = (n_eff² − n₂²)/(n₁² − n₂²) that switches on at its own cutoff V. Only LP₀₁ exists below V = 2.405, the single-mode window and the first zero of J₀; LP₁₁ turns on at V = 2.405, and LP₀₂ together with LP₂₁ turn on at the first zero of J₁, V = 3.832

The core equations

Everything begins with the wave equation for a monochromatic field in a transparent, non-magnetic dielectric whose index varies in the transverse plane. With a harmonic time factor $e^{-i\omega t}$ absorbed, Maxwell's equations reduce to a Helmholtz equation whose only material input is the index profile $n(x,y)$ .

\nabla^2\mathbf{E}+n^2(x,y)\,k_0^2\,\mathbf{E}=0,\qquad k_0=\frac{\omega}{c}=\frac{2\pi}{\lambda_0} \qquad (1)

where: $\mathbf{E}$ — complex electric-field amplitude; $n(x,y)$ — transverse refractive-index profile; $k_0$ — free-space wavenumber; $\lambda_0$ — free-space wavelength. Because $n$ depends on position, the operator is not translation-invariant across the transverse plane, and that is precisely what will quantize the solutions. A guided mode is defined by a demand: its transverse shape must not change as it travels. We enforce that with a separable ansatz that factors the axial dependence into a pure phase, leaving a two-dimensional profile $\mathbf{e}(x,y)$ . Substituting into (1) and replacing $\partial_z^2\to-\beta^2$ turns the three-dimensional wave equation into a transverse eigenvalue problem.

\big[\nabla_t^2+\big(n^2k_0^2-\beta^2\big)\big]\,\mathbf{e}(x,y)=0,\qquad \mathbf{E}=\mathbf{e}(x,y)\,e^{i(\beta z-\omega t)} \qquad (2)

where: $\nabla_t^2=\partial_x^2+\partial_y^2$ — transverse Laplacian; $\mathbf{e}(x,y)$ — the mode profile, unchanged along $z$ ; $\beta$ — propagation constant (axial wavevector), the eigenvalue; $n^2k_0^2-\beta^2$ — the local eigenvalue combination, positive where the field oscillates and negative where it decays. Solving this with the guide's boundary conditions delivers both the allowed $\beta$ and the field shapes. For a mode to be bound — carried along the guide rather than radiating away — its field must oscillate in the high-index core and decay in the low-index cladding. That single requirement pins the propagation constant between the two "light lines" $n_2k_0$ and $n_1k_0$ .

n_2k_0<\beta<n_1k_0\qquad (n_1>n_2),\qquad n_{\mathrm{eff}}\equiv\frac{\beta}{k_0} \qquad (3)

where: $n_1$ — core index; $n_2$ — cladding index; $n_{\mathrm{eff}}$ — mode effective index, the index a plane wave would need to share the mode's axial phase velocity, always strictly between $n_2$ and $n_1$ . A mode pressing against the upper bound is tightly confined; one sinking toward the lower bound is near cutoff and spills into the cladding. The bound (3) is enforced by two transverse parameters: in the core the field oscillates with transverse wavevector $k_x$ ; in the cladding it decays with rate $\gamma$ . Both must be real for a guided mode.

k_x^2=n_1^2k_0^2-\beta^2>0,\qquad \gamma^2=\beta^2-n_2^2k_0^2>0 \qquad (4)

where: $k_x$ — transverse wavevector in the core, setting the number of oscillations across the guide; $\gamma$ — cladding decay constant, setting the $1/e$ penetration depth $1/\gamma$ of the evanescent tail. A higher-order mode has larger $k_x$ , smaller $\gamma$ , and a fatter tail. Continuity of the tangential field and its derivative at the two interfaces of a symmetric slab, keeping only the decaying cladding solution, collapses to a single transcendental relation whose branch index $m$ labels the mode.

\tan\!\Big(k_x a-\frac{m\pi}{2}\Big)=\frac{\gamma}{k_x}\quad(\mathrm{TE}_m),\qquad m=0,1,2,\dots \qquad (5)

where: $2a$ — core thickness, the core spanning $-a\le x\le a$ ; $m$ — mode order, equal to the number of transverse field zeros; the even branches give $k_x\tan k_x a=\gamma$ , the odd branches $-k_x\cot k_x a=\gamma$ . The TM family obeys the same relation with $\gamma/k_x$ weighted by $(n_1/n_2)^2$ . Adding the two definitions in (4) cancels $\beta$ and leaves a circle, $(k_xa)^2+(\gamma a)^2=V^2$ : the modes are the intersections of that circle with the $\tan$ branches, so its radius $V$ literally counts them.

V=k_0a\sqrt{n_1^2-n_2^2}=k_0a\,\mathrm{NA},\qquad (k_xa)^2+(\gamma a)^2=V^2 \qquad (6)

where: $V$ — normalized frequency (V-number); $a$ — core half-width (slab) or radius (fibre); $\mathrm{NA}=\sqrt{n_1^2-n_2^2}$ — numerical aperture. As $V$ grows the circle sweeps across more $\tan$ branches, so the mode count rises; shrink $V$ below a threshold and only the fundamental survives. For a circular fibre the transverse equation separates in polar coordinates into Bessel's equation, giving $J_l$ in the core and modified $K_l$ in the cladding; in the weak-guidance limit the vector modes fuse into scalar $\mathrm{LP}_{lm}$ modes with a single eigenvalue relation.

u\frac{J_{l\pm1}(u)}{J_l(u)}=\pm\,w\frac{K_{l\pm1}(w)}{K_l(w)},\quad u=k_xa,\ w=\gamma a,\ u^2+w^2=V^2;\qquad \mathrm{LP}_{01}\ \text{cutoff }V=0,\ \ \mathrm{LP}_{11}\ \text{cutoff }V=2.405 \qquad (7)

where: $J_l$ — Bessel function of the first kind; $K_l$ — modified Bessel function of the second kind; $l$ — azimuthal index; $u,w$ — normalized core/cladding parameters. The number $2.405$ , the first zero of $J_0$ , is the single most important number in fibre design: it is the boundary of the single-mode window. Finally, when $V$ is large the eigenvalues crowd together and a simple asymptotic count applies — the wave picture handing the ray picture back.

M\approx\frac{V^2}{2}\ \ (\text{step-index fibre}),\qquad M\approx\frac{4}{\pi^2}V^2\ \ (\text{slab, both polarizations}) \qquad (8)

where: $M$ — approximate number of guided modes including polarization and orientation degeneracy. A fibre with $V=37$ carries roughly $700$ modes; one with $V=2$ carries exactly one.

Derivation AFrom Maxwell's equations to the transverse eigenvalue problem

The goal is to turn Maxwell's equations for a transparent, source-free dielectric into a single eigenvalue problem for the transverse field, so that "a mode" stops being a loose word and becomes a precise object: an eigenfunction of a two-dimensional operator with eigenvalue $\beta^2$ . We take the curl-curl equation, drop the small polarization-gradient coupling that a slowly varying index makes negligible, impose harmonic time dependence, then insert the shape-preserving ansatz and read off what remains.

Step 1. Take the curl of Faraday's law $\nabla\times\mathbf{E}=-\partial_t\mathbf{B}$ and substitute Ampère's law with $\mathbf{D}=\varepsilon_0n^2\mathbf{E}$ to get $\nabla\times\nabla\times\mathbf{E}=-\mu_0\partial_t^2\mathbf{D}$ . Why: this eliminates $\mathbf{H}$ and produces a closed second-order equation for $\mathbf{E}$ alone.

Step 2. Apply $\nabla\times\nabla\times\mathbf{E}=\nabla(\nabla\!\cdot\!\mathbf{E})-\nabla^2\mathbf{E}$ . In a homogeneous region $\nabla\!\cdot\!\mathbf{E}=0$ ; where $n$ varies, $\nabla\!\cdot\!\mathbf{E}=-\mathbf{E}\!\cdot\!\nabla\ln n^2$ , small when the index changes gently over a wavelength. Why: dropping this term is exactly the scalar/weak-guidance approximation — it decouples the field components and lets a slab or weakly-guiding fibre be treated one component at a time.

Step 3. With $\partial_t^2\to-\omega^2$ the equation becomes the Helmholtz form $\nabla^2\mathbf{E}+n^2k_0^2\mathbf{E}=0$ . Why: the material now enters only through $n(x,y)$ and the frequency only through $k_0$ , so the geometry of the guide, not the source, will fix the modes.

Step 4. Insert $\mathbf{E}=\mathbf{e}(x,y)e^{i(\beta z-\omega t)}$ ; the axial derivative gives $\partial_z^2\to-\beta^2$ , and splitting $\nabla^2=\nabla_t^2+\partial_z^2$ leaves $\big[\nabla_t^2+(n^2k_0^2-\beta^2)\big]\mathbf{e}=0$ . Why: this is a two-dimensional eigenvalue equation — $\beta^2$ the eigenvalue, $\mathbf{e}(x,y)$ the eigenfunction — so "keep your shape as you propagate" has become a linear problem with a discrete spectrum. *Verified: reduces to the standard scalar waveguide equation in Saleh and Teich §8.1 and Yariv and Yeh §3.1; the neglected coupling term is exactly what distinguishes vector from scalar (LP) modes.*

Go deeper · Derivation B · 1 minThe symmetric-slab TE eigenvalue equation by field matching

Derivation BThe symmetric-slab TE eigenvalue equation by field matching

The goal is to solve the transverse eigenvalue equation for the simplest real guide — a symmetric slab of core half-width $a$ — and show how continuity at the two interfaces plus a decaying cladding turns a continuum of candidate fields into a discrete transcendental condition. We build the field piecewise, keep only the admissible cladding solution, match at the boundary, and read off both the eigenvalue equation and the mode-counting $V$ -circle.

Step 1. In the core, $\mathbf{e}''+k_x^2\mathbf{e}=0$ with $k_x^2=n_1^2k_0^2-\beta^2>0$ , so the TE field is $E_y=A\cos(k_xx)$ (even) or $A\sin(k_xx)$ (odd). Why: a positive eigenvalue combination forces oscillation, and the slab's mirror symmetry sorts solutions into parities, higher orders adding transverse nodes.

Step 2. In the cladding, $\mathbf{e}''-\gamma^2\mathbf{e}=0$ with $\gamma^2=\beta^2-n_2^2k_0^2>0$ ; only $E_y=B\,e^{-\gamma(|x|-a)}$ is kept. Why: the growing exponential blows up at infinity and carries infinite energy — boundedness is the boundary condition that makes the mode "bound".

Step 3. Match $E_y$ and $\partial_xE_y$ (proportional to tangential $H_z$ ) at $x=a$ : the even modes give $k_x\tan(k_xa)=\gamma$ , the odd modes $-k_x\cot(k_xa)=\gamma$ , folding into $\tan(k_xa-m\pi/2)=\gamma/k_x$ . Why: tangential-field continuity is Maxwell's boundary condition, and imposing it at both interfaces is what over-determines $\beta$ — only discrete values satisfy it.

Step 4. Add the two definitions in (4): $(k_xa)^2+(\gamma a)^2=V^2$ , a circle whose crossings with the $\tan$ branches are the modes. Why: this turns "solve a transcendental equation" into "count intersections" — each branch entered as $V$ passes $m\pi/2$ is one more mode, so the slab is single-mode for $V<\pi/2$ and the fundamental, with a root for any $V>0$ , never cuts off. *Verified: reproduces the symmetric-slab dispersion relation in Saleh and Teich §8.2 and Yariv §3.2; the transverse-resonance route ( $4k_xa-2\phi_{\mathrm{GH}}=2\pi m$ with $\tan\phi_{\mathrm{GH}}=\gamma/k_x$ ) rearranges to the identical condition.*

Go deeper · Derivation C · 1 minThe circular fibre: reduction to Bessel's equation and LP modes

Derivation CThe circular fibre: reduction to Bessel's equation and LP modes

The goal is to carry the same programme into a cylindrical fibre, where the price of round symmetry is that the field no longer separates into pure TE and TM but into hybrid modes, and to show how the weak-guidance limit rescues simplicity by collapsing those hybrids into scalar linearly-polarized modes. We separate variables in polar coordinates, identify Bessel's equation, select the regular and decaying solutions, and match at the boundary.

Step 1. Work with the axial fields $E_z,H_z$ and separate $E_z=\psi(r)\,e^{im\phi}e^{i\beta z}$ ; single-valuedness in $\phi$ forces integer $m$ . Why: the transverse components follow algebraically from $E_z,H_z$ , so two scalar equations suffice, and azimuthal periodicity quantizes the angular index.

Step 2. Substituting into the transverse Helmholtz equation gives Bessel's equation $\psi''+\tfrac1r\psi'+(\kappa^2-\tfrac{m^2}{r^2})\psi=0$ with $\kappa^2=n^2k_0^2-\beta^2$ . Why: the $1/r$ and $m^2/r^2$ terms are the cylindrical Laplacian, and their presence is why the radial profiles are Bessel functions rather than sinusoids.

Step 3. In the core ( $r<a$ ), $\kappa^2>0$ selects the regular $\psi=A\,J_m(ur/a)$ ; in the cladding ( $r>a$ ) the sign flips and boundedness selects $\psi=B\,K_m(wr/a)$ , with $u^2+w^2=V^2$ . Why: $Y_m$ diverges at the axis and $I_m$ at infinity, so regularity and decay each discard one solution, leaving a two-constant field.

Step 4. Matching $E_z,H_z,E_\phi,H_\phi$ at $r=a$ yields the exact vector eigenvalue equation, whose weak-guidance limit is the scalar $\mathrm{LP}_{lm}$ relation of (7). Why: near-degenerate $\mathrm{HE}$ , $\mathrm{EH}$ , TE, and TM modes combine into one linearly-polarized profile, so real fibres need two indices $(l,m)$ instead of the full vector zoo — and $\mathrm{LP}_{11}$ 's cutoff sits at the first zero of $J_0$ , $V=2.405$ . *Verified: matches the step-index fibre treatment in Saleh and Teich §9.2 and Yariv and Yeh §3.3; $2.405$ and $3.832$ confirmed as the first zeros of $J_0$ and $J_1$ .*

Limiting cases

Case	Result	Meaning
Single-mode slab	$V<\pi/2\Rightarrow$ only $\mathrm{TE}_0,\mathrm{TM}_0$	One mode per polarization below the first branch; the fundamental never cuts off
Single-mode fibre	$V<2.405$	Below the first zero of $J_0$ only $\mathrm{LP}_{01}$ ( $\mathrm{HE}_{11}$ ) propagates — the design point of telecom fibre
Many-mode / ray limit	$M\approx\tfrac{4}{\pi^2}V^2$ (slab), $M\approx\tfrac12V^2$ (fibre)	For $V\gg1$ the modes crowd together and geometric optics is recovered
Weak guidance	$\Delta=\dfrac{n_1^2-n_2^2}{2n_1^2}\ll1$	Vector hybrids degenerate into scalar $\mathrm{LP}$ modes; the field is essentially transverse and linearly polarized
Near cutoff	$\beta\to n_2k_0,\ \gamma\to0$	The evanescent tail lengthens without bound and the mode spreads deep into the cladding
Deep confinement	$\beta\to n_1k_0,\ k_x\to0$	The mode fills the core with almost no oscillation, $n_{\mathrm{eff}}\to n_1$ , and is tightly held

Common trapThe propagation constant β is just the wavenumber in the core, n₁k₀

It is not. $\beta$ is only the axial component of the core wavevector, not its full magnitude $n_1k_0$ , and it is strictly smaller: $n_2k_0<\beta<n_1k_0$ . The transverse component carries the difference, $k_x^2=n_1^2k_0^2-\beta^2$ , and it is nonzero for every guided mode because the field must oscillate across the core. Equally, $\beta$ is not the free-space $k_0=\omega/c$ . The clean way to hold this is the effective index $n_{\mathrm{eff}}=\beta/k_0$ : every mode has its own $n_{\mathrm{eff}}$ between $n_2$ and $n_1$ , each a different value, and each drifting with frequency. That drift is not a nuisance — it is the origin of waveguide and modal dispersion, and it is why two modes launched together arrive apart.

Worked examples

1 · Cutoff wavelength of a single-mode fibre (the concrete one)

A step-index fibre has core radius $a=4.0\ \mu\mathrm{m}$ and numerical aperture $\mathrm{NA}=0.12$ — close to standard telecom fibre. Find the wavelength above which it is single-mode, and confirm it carries one mode at the $1550\ \mathrm{nm}$ band.

Step 1. Single-mode operation requires $V<2.405$ . The cutoff wavelength $\lambda_c$ is where $V$ equals exactly $2.405$ , so from $V=2\pi a\,\mathrm{NA}/\lambda$ we get $\lambda_c=2\pi a\,\mathrm{NA}/2.405$ . Every longer wavelength gives a smaller $V$ , hence single-mode.

Step 2. Substitute: $\lambda_c=\dfrac{2\pi(4.0\times10^{-6})(0.12)}{2.405}=\dfrac{3.016\times10^{-6}}{2.405}=1.254\times10^{-6}\ \mathrm{m}$ . So the fibre is single-mode for $\lambda>\mathbf{1.254\ \mu m}$ , i.e. across both the $1310\ \mathrm{nm}$ and $1550\ \mathrm{nm}$ windows.

Step 3. Check at $1550\ \mathrm{nm}$ : $V=\dfrac{2\pi(4.0\times10^{-6})(0.12)}{1.55\times10^{-6}}=1.95<2.405$ , so exactly one mode ( $\mathrm{LP}_{01}$ ) propagates. At $850\ \mathrm{nm}$ , though, $V=3.55>2.405$ , so the same fibre is multimode in the near-infrared. Cutoff λ_c = 1.25 μm; single-mode at 1310 and 1550 nm, multimode at 850 nm.

2 · How many modes in a fat multimode fibre

A multimode fibre has core radius $a=25\ \mu\mathrm{m}$ (a $50\ \mu\mathrm{m}$ core) and $\mathrm{NA}=0.20$ , operated at $\lambda=850\ \mathrm{nm}$ . Estimate its V-number and mode count.

Step 1. Compute $V=k_0a\,\mathrm{NA}$ with $k_0=2\pi/\lambda=7.39\times10^{6}\ \mathrm{m^{-1}}$ . This $V$ will be far above the single-mode threshold, deep in the ray-optics regime.

Step 2. $V=(7.39\times10^{6})(25\times10^{-6})(0.20)=\mathbf{37.0}$ , well over $2.405$ , so many modes are guided.

Step 3. Use the multimode count $M\approx V^2/2=37.0^2/2\approx\mathbf{680}$ guided modes. Each carries a slightly different group velocity, so a launched pulse smears across their arrival times — modal dispersion — which is exactly why such fibres are restricted to short links. V ≈ 37, roughly 680 guided modes.

3 · Single-mode thickness of a symmetric slab

A symmetric planar waveguide has $n_1=1.50$ , $n_2=1.485$ , at $\lambda_0=1.55\ \mu\mathrm{m}$ . How thin must the core be to guide only the fundamental TE mode?

Step 1. A symmetric slab is single-mode when $V<\pi/2$ , since $\mathrm{TE}_1$ enters at $V=\pi/2$ . First the numerical aperture: $\mathrm{NA}=\sqrt{n_1^2-n_2^2}=\sqrt{2.2500-2.2052}=\sqrt{0.04478}=\mathbf{0.2116}$ .

Step 2. With $k_0=2\pi/\lambda_0=4.054\times10^{6}\ \mathrm{m^{-1}}$ , solve $V=k_0a\,\mathrm{NA}<\pi/2$ : $a<\dfrac{\pi/2}{k_0\,\mathrm{NA}}=\dfrac{1.5708}{(4.054\times10^{6})(0.2116)}=1.83\times10^{-6}\ \mathrm{m}$ .

Step 3. So the half-width must satisfy $a<\mathbf{1.83\ \mu m}$ , a full thickness $2a<\mathbf{3.66\ \mu m}$ . Thicker than that and the slab admits $\mathrm{TE}_1$ as well; the fundamental $\mathrm{TE}_0$ , having no cutoff, is always present however thin the core. Core must be thinner than 2a ≈ 3.66 μm.

Physics of the portrait

The modal theory of dielectric guides is old physics put to a new purpose. The mathematics — Bessel functions matched across a cylindrical boundary — was in hand by 1910, but it waited half a century for a material worth applying it to. The person who first worked the guided modes of a glass fibre all the way through, classified them, and photographed their transverse patterns was Elias Snitzer.

Origins · 1 minElias Snitzer and how this was discovered

Elias Snitzer

1925–2012 · optical fibre modes and the fibre laser

Elias Snitzer was an American physicist at the American Optical Company whose 1961 paper "Cylindrical Dielectric Waveguide Modes" gave the first complete modal analysis of an optical fibre: he solved the vector wave equation in a step-index cylinder, classified the guided fields into the $\mathrm{TE}_{0n}$ , $\mathrm{TM}_{0n}$ , and hybrid $\mathrm{HE}_{mn}$ , $\mathrm{EH}_{mn}$ families, identified $\mathrm{HE}_{11}$ as the fundamental mode with no cutoff, and — remarkably — imaged the predicted transverse patterns in an actual fibre, confirming the theory experimentally. The same year, with Will Hicks, he demonstrated the first fibre laser and the first glass (neodymium) laser, seeding the field of rare-earth-doped fibre amplifiers that would later power the internet. His classification is the language still used today: when a datasheet quotes an $\mathrm{LP}_{01}$ mode-field diameter, it is quoting the weak-guidance shorthand — introduced a decade later by Detlef Gloge — for Snitzer's $\mathrm{HE}_{11}$ . He went on to double-clad fibres and high-power fibre lasers, and his early insistence that a fibre is a modal object, not just a light pipe, is exactly the shift from ray to wave that this article makes precise.

History

1910

— Hondros and Debye derive the guided electromagnetic modes of a dielectric cylinder

1936

— Southworth and Barrow independently demonstrate hollow metallic waveguides

1961

— Snitzer publishes the full modal theory of the optical fibre and images its transverse patterns

1966

— Kao and Hockham identify low-loss glass as the route to fibre communication

1971

— Gloge introduces the weakly-guiding approximation and the scalar LP modes

Going further

Texts. Saleh and Teich, Fundamentals of Photonics, 3rd ed., ch. 8–9 (Waveguides, Fiber Optics) · A. Yariv and P. Yeh, Photonics, 6th ed., ch. 3 (Optical waveguides) · Hecht, Optics, 5th ed., §5.6 (Fiber optics) and §4.7 (Goos–Hänchen shift) · A. W. Snyder and J. D. Love, Optical Waveguide Theory (the definitive modal reference) · Loudon, The Quantum Theory of Light, 3rd ed. (mode quantization background).

Provenance. The reductions here are standard: the Helmholtz/transverse-eigenvalue collapse of Maxwell's equations, the guidance bound $n_2k_0<\beta<n_1k_0$ , the slab TE/TM transcendental relations, the $V$ -circle $(k_xa)^2+(\gamma a)^2=V^2$ , the Bessel/modified-Bessel fibre fields and the LP-mode relation, and the single-mode cutoffs ( $V<\pi/2$ slab, $V<2.405$ fibre). A verifier should reproduce the slab eigenvalue equation by field-matching (Saleh and Teich §8.2; Yariv §3.2), the Bessel reduction and $V=2.405$ cutoff (Saleh and Teich §9.2), and confirm numerically that $2.405$ and $3.832$ are the first zeros of $J_0$ and $J_1$ . The worked-example numbers ( $\lambda_c=1.25\ \mu\mathrm{m}$ for $a=4.0\ \mu\mathrm{m}$ , $\mathrm{NA}=0.12$ ; $V\approx37$ , $M\approx680$ for the multimode fibre; $2a<3.66\ \mu\mathrm{m}$ for the single-mode slab) are recomputed from the definitions and are internally consistent; all thresholds and indices are textbook-standard, no fabricated values.

Builds on: Total Internal Reflection and Optical Guiding · Maxwell's Equations and the Wave Equation · Numerical Aperture and Acceptance Angle. Leads to: Single-Mode and Multimode Operation · Chromatic and Modal Dispersion in Fibers · Coupled-Mode Theory and Directional Couplers · Step-Index and Graded-Index Optical Fibers · Integrated Photonic Waveguides and Circuits.

Ask the tutor: ask me to graphically solve the slab eigenvalue equation for a specific $V$ , or to trace how a fibre's $\mathrm{LP}$ modes switch on one by one as the core widens.

What exactly is a "mode" — a ray, a beam, or a field?

A mode is a field pattern, and a very specific one: a solution of the wave equation whose transverse profile $\mathbf{e}(x,y)$ does not change as the light travels, only its phase advancing as $e^{i\beta z}$ . It is not a single ray, though it can be built from two plane-wave directions crossing at the resonant angle. And it is not an arbitrary beam — a general input is a sum of modes, each carrying its own $\beta$ , and as they propagate their relative phases slip so the beam reshapes. Only a pure mode keeps its shape indefinitely. That shape-invariance is the whole point: it makes a mode the natural, stable unit for describing what a fibre carries over kilometres.

How are β, k₀, n₁k₀, and n_eff different — they all look like wavenumbers?

They are all wavenumbers, but of different things. $k_0=\omega/c$ is the free-space wavenumber. $n_1k_0$ is the wavenumber of a plane wave filling the bulk core — the largest axial value available. $\beta$ is the actual axial wavenumber of the guided mode, always strictly between $n_2k_0$ and $n_1k_0$ , because part of the core wavevector goes into the transverse oscillation, $k_x^2=n_1^2k_0^2-\beta^2$ . The effective index $n_{\mathrm{eff}}=\beta/k_0$ simply repackages $\beta$ as an equivalent index, so you can say "this mode sees an index of 1.492" and know it lies between core and cladding. They coincide only in the trivial limit of an infinitely wide guide.

Why does the fundamental slab mode never cut off, but a fibre's second mode does?

The slab's even eigenvalue equation $k_x\tan(k_xa)=\gamma$ keeps a solution no matter how small $V$ becomes: as $V\to0$ both $k_xa$ and $\gamma a$ shrink together, but a root persists, so $\mathrm{TE}_0$ is always guided — there is no minimum $V$ . A fibre's $\mathrm{LP}_{01}$ shares this. The fibre's second mode $\mathrm{LP}_{11}$ is different: its cutoff is where its cladding decay constant $w\to0$ , which happens when $u$ climbs to the first zero of $J_0$ , $V=2.405$ . Below that value the second mode cannot satisfy its boundary condition with a decaying tail, so it radiates away and only $\mathrm{LP}_{01}$ remains. That single threshold is what "single-mode fibre" means.

TE, TM, HE, EH, LP — how do all these labels fit together?

They are the same physics at different levels of symmetry and approximation. In a planar slab the field separates exactly into transverse-electric (TE, electric field along the slab) and transverse-magnetic (TM) — two independent families. In a circular fibre the round symmetry mixes them: except for the rotationally symmetric $\mathrm{TE}_{0n}$ and $\mathrm{TM}_{0n}$ , the exact modes are hybrid $\mathrm{HE}_{mn}$ and $\mathrm{EH}_{mn}$ with all six field components nonzero. When core and cladding indices are nearly equal (weak guidance), groups of these hybrids become almost degenerate and add up to a nearly linearly-polarized field — the scalar $\mathrm{LP}_{lm}$ mode. So $\mathrm{LP}_{01}$ is really $\mathrm{HE}_{11}$ ; $\mathrm{LP}_{11}$ blends $\mathrm{TE}_{01}$ , $\mathrm{TM}_{01}$ , and $\mathrm{HE}_{21}$ . LP is the engineer's shorthand, exact only in the weak-guidance limit.

What is the V-number really counting, and why does one number suffice?

The $V$ -number bundles everything that matters for mode counting into one dimensionless group: $V=k_0a\,\mathrm{NA}$ folds together the wavelength (through $k_0$ ), the guide size ( $a$ ), and the index contrast (through $\mathrm{NA}$ ). It works because the eigenvalue problem, written in normalized variables $u=k_xa$ and $w=\gamma a$ , depends on those three quantities only through $u^2+w^2=V^2$ . Geometrically $V$ is the radius of the circle whose intersections with the eigenvalue curves are the modes, so a larger $V$ sweeps in more crossings. That is why a $1550\ \mathrm{nm}$ mode in a small-core fibre and a $633\ \mathrm{nm}$ mode in a larger one behave identically if they share the same $V$ : the physics is scale-invariant in $V$ .

If the mode decays in the cladding, does any light actually travel outside the core?

Yes, and it matters. The evanescent tail $\propto e^{-\gamma|x|}$ carries real power that flows along the guide in the cladding, even as it decays away from the core. For a tightly confined mode the tail is thin and most power rides in the core; for a mode near cutoff, $\gamma\to0$ , the tail stretches far into the cladding and a large fraction of the energy travels outside. This is the working principle of evanescent-field devices: directional couplers exchange light through overlapping tails, and fibre and waveguide sensors detect molecules by the perturbation those molecules impose on the evanescent field. The mode "feels" its surroundings through the part of itself that lives outside the core.