Total Internal Reflection and Optical Guiding

The most direct modern heir of the evanescent field is total-internal-reflection fluorescence (TIRF) microscopy. A laser enters a glass coverslip above the critical angle, and the roughly 100 nm evanescent skin — precisely the $d_p=1/\kappa$ above — illuminates only the sliver of a cell pressed against the glass, leaving the interior dark. That thin-sheet excitation gives the background-free contrast single-molecule biophysics needs to watch motor proteins step at the membrane. The same tail, read as a phase shift, lets surface-plasmon-resonance and attenuated-total-reflection sensors detect a monolayer of binding analyte without a label. At the other extreme of scale, repeated TIR is the substrate of the optical internet. Submarine fibre relies on the losslessness proved in Derivation 1: with real silica indices the bounce returns essentially all the power, so attenuation falls to a few tenths of a dB per kilometre near 1550 nm. On-chip, silicon photonics pushes the contrast the other way — a silicon core near $3.5$ in oxide near $1.45$ gives a critical angle small enough to bend light around micron-scale corners, packing modulators and microrings onto one die. And in atomic physics, a blue-detuned evanescent field at a glass surface builds a repulsive, near-lossless "atom mirror" whose steepness is set by $\kappa$, bouncing cold atoms without touching them.

Take a ray inside a block of glass and swing it toward the top surface. At modest angles it splits into a refracted ray that bends away from the normal into the air and a reflected ray back down. Tilt it further and the escaping ray leans harder toward the surface, until at one angle it lies flat — grazing, going nowhere. A hair more and the surface hands all the light back. That tipping point is the critical angle $\theta_c$, and beyond it the interface is an almost perfect mirror that cost nothing to build — no coating, just two transparent media. This is why a cut diamond blazes: its index is so high that $\theta_c\approx24^\circ$, so most rays that get inside are trapped and shuttled to the top facets before escaping to your eye.

The reflected wave is not quite what a metal mirror gives back. It is phase-delayed, behaving as though it bounced from a plane just beyond the interface — the Goos–Hänchen shift. The carrier of that delay is the evanescent field, a real oscillating field that soaks a fraction of a wavelength into the rarer medium and rides along the surface without carrying energy across. That field is no fiction — bring a second high-index medium within its reach and it pulls power across the gap, the mechanism behind frustrated TIR, directional couplers, and evanescent sensors. Guiding is simply TIR made a habit. Sandwich a high-index core between lower-index cladding and any ray steep enough to exceed $\theta_c$ at the wall is trapped, marching down the guide as a shallow zig-zag that never leaks. But not every angle survives. Because each bounce stamps the wave with a phase, only rays whose transverse round-trip phase closes on itself — a standing pattern across the core — persist. That self-consistency turns the continuum of trapped angles into a discrete set of guided modes, where this ray story meets the wave equation of aq-optics-waveguide-modes-wave-equation.

Snell's law is the refraction bookkeeping at any interface, and reading it from the dense side is what exposes the critical angle: the product of index and the sine of the angle from the normal is conserved across the boundary.

where: $n_1,n_2$ — refractive indices of the incidence (core) and transmission (cladding) media; $\theta_1,\theta_2$ — angles of incidence and refraction from the surface normal. Physical reading: when $n_1>n_2$ the refracted ray bends away from the normal, so as $\theta_1$ grows, $\theta_2$ reaches $90^\circ$ first — the sign that refraction is about to fail. Set the refracted ray to graze the surface, $\theta_2=90^\circ$, and Snell's law fixes the incidence angle at which transmission shuts off; for any larger angle there is no real $\theta_2$, and TIR takes over.

where: $\theta_c$ — critical angle; TIR occurs for every $\theta_1>\theta_c$. Physical reading: a real $\theta_c$ exists only for $n_2/n_1<1$, so TIR is strictly dense-to-rare. Large contrast shrinks $\theta_c$ (glass in air, $41.8^\circ$); small contrast pushes it toward $90^\circ$ (silica core and cladding, $80.6^\circ$), trapping only near-axial rays. Beyond $\theta_c$ the transmitted field survives as an evanescent wave. Continuity of the tangential wavevector fixes the in-plane component; the medium-2 dispersion relation then forces the normal component imaginary — an exponential decay set by $\kappa$, with penetration depth $d_p=1/\kappa$.

where: $k_0=2\pi/\lambda_0$ — free-space wavenumber; $\kappa$ — amplitude decay constant into the rarer medium; $d_p$ — penetration depth. Physical reading: well above $\theta_c$ the field dies within a fraction of $\lambda_0$; as $\theta_1\to\theta_c^{+}$, $\kappa\to0$ and $d_p$ diverges — grazing modes near cutoff reach deep into the cladding and are most vulnerable to bend and cladding loss. The Fresnel coefficient carries the whole "lossless but phase-shifting" story. Past $\theta_c$ the transmission-angle cosine turns imaginary, so the TE coefficient becomes a ratio of a complex number to its own conjugate — unit modulus, pure phase.

where: $r_s$ — amplitude reflection coefficient for s (TE) polarization; $\phi_s$ — phase acquired on total reflection. Physical reading: $R=|r_s|^2=1$ says all power returns, while $\phi_s$ runs from $0$ at $\theta_c$ toward $-\pi$ at grazing — the differing p-phase is why TIR converts linear into elliptical polarization. Guiding closes the loop by demanding self-consistency: a ray that crosses the core, bounces, and returns must reproduce its own transverse phase or interfere itself away. The path phase plus the two bounce phases must total a multiple of $2\pi$, selecting discrete angles labelled by an integer $m$.

where: $d$ — core thickness (slab) or diameter scale; $\phi(\theta_1)$ — TIR phase per bounce; $m$ — mode number. Physical reading: each integer $m$ keeping $\theta_1>\theta_c$ is one guided mode. A thin core or small contrast admits only $m=0$ (single-mode); a thick, high-contrast guide admits many. Finally, the critical angle at the wall reaches back to the input face and fixes how wide a cone of external light can be launched. Combining the guiding condition with refraction at the entrance face gives the numerical aperture.

where: $\theta_a$ — maximum external acceptance half-angle; $n_0$ — index of the launch medium (air, $n_0=1$). Physical reading: NA depends only on the two indices, not the core size; large contrast gathers a wide cone (multimode), a weakly guiding fibre only near-axial rays (small NA, often single-mode).

How the interface behaves is governed mostly by where the incidence angle sits relative to $\theta_c$, and secondarily by the index contrast, which sets both $\theta_c$ and the numerical aperture.

1 · Critical angle, numerical aperture, and acceptance cone of a step-index fibre (confidence builder)

A step-index fibre has core $n_1=1.500$ and cladding $n_2=1.480$, launched from air ($n_0=1$). Find the critical angle at the wall, the numerical aperture, and the acceptance cone half-angle.

Step 1. From equation (2), $\theta_c=\arcsin(1.480/1.500)=\arcsin(0.98667)=\mathbf{80.6^\circ}$ — only $9.4^\circ$ shy of grazing. A guided ray must strike the wall steeper than this from the normal, i.e. within $90^\circ-\theta_c=9.4^\circ$ of the fibre axis.

Step 2. The numerical aperture, equation (6), depends only on the indices: $\mathrm{NA}=\sqrt{1.500^2-1.480^2}=\sqrt{0.0596}=\mathbf{0.244}$. The small value flags a weakly guiding fibre — the near-equal indices gather only a narrow cone.

Step 3. The external acceptance half-angle is $\theta_a=\arcsin(\mathrm{NA}/n_0)=\arcsin(0.244)=\mathbf{14.1^\circ}$. Check: the steepest internal ray sits $9.4^\circ$ from the axis, and refracting it out gives $n_1\sin(9.4^\circ)=0.245$, matching the NA. Light must arrive within a 14.1° half-angle cone in air to be trapped; anything wider leaks into the cladding.

2 · How deep does the evanescent field reach, and by how much is the wave phase-shifted?

A HeNe beam ($\lambda_0=633\ \mathrm{nm}$) undergoes TIR at a glass–air interface ($n_1=1.50$, $n_2=1.00$) at $\theta_1=45^\circ$. Find the penetration depth $d_p$ and the TE reflection phase $\phi_s$.

Step 1. Confirm TIR: $\theta_c=\arcsin(1.00/1.50)=41.8^\circ<45^\circ$. The radical is $\sqrt{n_1^2\sin^2\theta_1-n_2^2}=\sqrt{2.25\times0.5-1}=\sqrt{0.125}=0.3536$.

Step 2. From equation (3), $d_p=\lambda_0/\left(2\pi\sqrt{n_1^2\sin^2\theta_1-n_2^2}\right)=633/(2\pi\times0.3536)=\mathbf{285\ nm}$, about $0.45\,\lambda_0$. The field is essentially gone within half a wavelength — close enough that a second glass surface within a few hundred nanometres would frustrate the reflection and let light tunnel across.

Step 3. The TE reflection phase, equation (4): with $\sin^2\theta_c=(1/1.5)^2=0.4444$ and $\sin^2 45^\circ=0.5$, the argument is $\sqrt{0.5-0.4444}/\cos45^\circ=0.3333$, so $\phi_s=-2\arctan(0.3333)=\mathbf{-36.9^\circ}$ — nonzero and angle-dependent, unlike a metal mirror's fixed shift, and (because the p-phase differs) what lets a Fresnel rhomb make circular polarization. The evanescent field reaches ≈ 285 nm into the air and the wave returns with a −36.9° phase.

3 · Single-mode versus multimode: the V-number

For the fibre of example 1 ($\mathrm{NA}=0.244$) at $\lambda_0=1.30\ \mu\mathrm{m}$, find the largest single-mode core radius, and the mode count if instead $a=25\ \mu\mathrm{m}$.

Step 1. A step-index fibre is single-mode for $V=2\pi a\,\mathrm{NA}/\lambda_0\le2.405$ (the first zero of $J_0$), so $a_{\max}=2.405\,\lambda_0/(2\pi\,\mathrm{NA})=2.405\times1.30/(2\pi\times0.244)=\mathbf{2.04\ \mu m}$, a core diameter near $4.1\ \mu\mathrm{m}$.

Step 2. With $a=25\ \mu\mathrm{m}$, $V=29.5$, and by Derivation 2 the mode count is $M\approx V^2/2\approx\mathbf{435\ modes}$. So this high-NA fibre needs a tiny core to stay single-mode; real telecom fibre uses a smaller NA ($\approx0.14$) so the core can be a manufacturable $\sim8\ \mu\mathrm{m}$. Single-mode operation trades index contrast against core size — high NA forces a tiny core.

Total internal reflection is one of optics' oldest observations and its guiding application one of the most consequential — yet the two were separated by more than a century, and the first person to bend a beam of light along a curving path is not the name most textbooks remember.

Texts. Hecht, Optics, 5th ed., §4.7 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 1 and Ch. 9 (waveguides) · Born & Wolf, Principles of Optics, §1.5 (evanescent waves) · Foot, Atomic Physics (atom mirrors) · Loudon, The Quantum Theory of Light · Demtröder, Laser Spectroscopy (fibre and waveguide sections).

Provenance. Check the critical angle, unit-modulus Fresnel coefficient, decay constant, and reflection phase against Hecht §4.7 and Saleh & Teich Ch. 6; the mode condition and numerical aperture against Saleh & Teich Ch. 9. A verifier should recompute $\theta_c=\arcsin(n_2/n_1)$, confirm $\lvert r_s\rvert=\lvert r_p\rvert=1$ above $\theta_c$, verify $\kappa=k_0\sqrt{n_1^2\sin^2\theta_1-n_2^2}$, and check $\mathrm{NA}=\sqrt{n_1^2-n_2^2}$ with the cutoff $V\le2.405$. The worked numbers ($\theta_c=80.6^\circ$, $\mathrm{NA}=0.244$; $d_p=285\ \mathrm{nm}$, $\phi_s=-36.9^\circ$; $\sim435$ modes) were recomputed here; fibre loss ($\sim0.2$ dB/km at 1550 nm) and silicon/silica indices ($3.5/1.45$) are standard, and the Colladon 1841 priority over Tyndall 1854 is documented.

Builds on: Snell's Law and Refraction · Fresnel Equations · Plane-Wave Solutions of Maxwell's Equations. Leads to: Numerical Aperture and Acceptance Angle · Waveguide Modes and the Wave Equation · Step-Index and Graded-Index Optical Fibres · Coupled-Mode Theory and Directional Couplers · Fibre Attenuation and Loss Mechanisms.

Ask the tutor: Ask me to walk through the transverse-resonance condition for a specific fibre and watch the discrete mode angles fall out of the phase-closure requirement.