Brewster's Angle and Total Internal Reflection

The Fresnel curves of the previous article carry two singular features, and each one built an industry. The first is a zero: at one angle, glass reflects absolutely none of the light polarized in the plane of incidence — not little, none — a fact David Brewster distilled in 1815 into the cleanest law in optics, $\tan\theta_B = n_2/n_1$ , and which today enforces the polarization of essentially every gas laser on Earth. The second is a wall: send light from the dense side toward the rare side steeply enough and Snell's law runs out of real angles. The interface stops transmitting altogether, reflecting with a fidelity no metal mirror approaches — the total internal reflection that pins light inside the optical fibers carrying this sentence to you. Both features live entirely inside the Fresnel coefficients already derived; this article adds no new boundary physics, only the two special geometries and what they contain. And they contain more than the headlines. The Brewster zero has a microscopic explanation of textbook-perfect elegance — the transmitted medium's dipoles are asked to radiate along their own axes, and dipoles cannot — plus a quieter geometric identity, reflected ⊥ refracted, that makes the formula memorable for life. Total internal reflection hides the stranger prize: the field in the "forbidden" medium is not zero but evanescent, clinging to the surface in a skin thinner than a wavelength, carrying no net power yet real enough to touch. Frustrate it with a second surface and light tunnels across a gap it cannot classically cross; interrogate it and you have ATR spectroscopy and fingerprint sensors; track its phases and a block of plain glass becomes an achromatic wave plate. Two angles, most of applied optics.

Key takeaways

Brewster's angle: $\tan\theta_B = n_2/n_1$ is where the Fresnel $r_p$ (previous article's coefficient, recalled not re-derived) passes through zero — reflected light is 100% $s$ -polarized. Geometry: reflected and refracted rays exactly perpendicular. Microscopics: dipoles in medium 2 cannot radiate along their own oscillation axis.
Brewster exists in both crossing directions ( $\theta_B$ and its internal partner are complementary: 56.3° and 33.7° for glass); it suppresses only the $p$ -channel — no $s$ -Brewster exists for nonmagnetic media.
Critical angle: for $n_1 > n_2$ , beyond $\sin\theta_c = n_2/n_1$ Snell's law has no real refraction angle; $\cos\theta_T$ turns imaginary and $|r| = 1$ in both channels — total internal reflection (glass→air 41.8°, water→air 48.8°).
The transmitted field above $\theta_c$ is evanescent: $E \propto e^{-\kappa z}$ with penetration depth $1/\kappa$ of order 100–600 nm for visible light, oscillating along the surface, carrying zero time-averaged power into medium 2.
TIR reflects with polarization-dependent phases ( $|r| = 1$ , $\arg r \neq 0$ ): the Fresnel rhomb stacks two internal reflections at 54.6° into an achromatic 90° retarder; a nearby second interface within the evanescent skin lets light tunnel across (frustrated TIR).
KEY IDEA: the two landmark angles are the Fresnel coefficients' singular geometries — the $p$ -mismatch crossing zero at $\tan\theta_B = n_2/n_1$ , and the refraction angle running out of real values at $\sin\theta_c = n_2/n_1$ , where transmission turns evanescent and reflection turns total.

The core insight

Brewster's angle ( $\tan\theta_B = n_2/n_1$ ) is where the parallel-polarization reflection coefficient vanishes, making reflected light fully $s$ -polarized; the critical angle ( $\sin\theta_c = n_2/n_1$ , dense-to-rare only) is where the transmitted wave turns evanescent and reflection becomes total — both exact consequences of the Fresnel coefficients plus Snell's law.

Where this lives today

Total internal reflection is arguably the most economically consequential piece of boundary-value physics ever solved: every optical fiber is a TIR machine (dressed in the waveguide-mode language this chapter builds next), and the evanescent skin is now an instrument. ATR-FTIR spectrometers press samples against a high-index crystal and read absorption directly out of the frustrated evanescent field — the standard quick chemical fingerprint in every analytical lab. TIRF microscopy illuminates only the ~100 nm evanescent slice above a coverslip, imaging single molecules at a cell's membrane against a dark bulk. Evanescent biosensors (surface plasmon resonance instruments among them) watch binding events shift the conditions of that skin in real time. The Brewster side is quieter but everywhere: laser Brewster windows and Brewster-angle cavity elements enforce polarization by pure geometry; Brewster-angle microscopy images single-molecule films on water by sitting exactly at the zero and watching anything that perturbs it light up; photography's polarizing filters and polarized sunglasses monetize the $R_s/R_p$ asymmetry daily. And the deepest modern echo of this article is in photonic design: the TIR phase shifts of Derivation C, generalized, are the boundary conditions that set the mode structure of integrated waveguides — the "phase budget" of every silicon-photonics chip starts from Fresnel's conjugate-pair arithmetic.

Origins · 1 minJohn Tyndall and how this was discovered

John Tyndall

1820–1893 · Royal Institution, London

Tyndall was an Irish surveyor's son who worked railway construction, taught school to fund a Marburg doctorate, and rose to succeed Faraday at the Royal Institution — the Victorian era's great scientific showman-experimentalist. In an 1854 Royal Institution lecture he bent a beam of light inside an arc of falling water: light poured from a tank through a nozzle and stayed inside the curving jet, guided bounce by bounce by total internal reflection (Swiss engineer Daniel Colladon had staged the same "light fountain" in 1841, a priority Tyndall's fame long obscured). That demonstration is the direct conceptual ancestor of the optical fiber — Charles Kao's 2009 Nobel citation traces the lineage. Tyndall's restless experiments ranged wider: he explained why the sky is blue via scattering from suspended particles (the Tyndall effect), made the first careful measurements of infrared absorption by water vapor and CO₂ — founding experimental greenhouse-gas science in 1859 — and effectively invented the firefighter's respirator. A mountaineer of note, he made the first ascent of the Weisshorn. He died of an accidental overdose administered by his wife, who survived him by 47 years and never finished the biography she withheld his papers to write.

History

1611

— Kepler describes total reflection in his Dioptrice, decades before Snell's law is published

1808

— Malus discovers polarization by reflection in sunset light off the Luxembourg Palace windows

1815

— Brewster establishes tan θB = n2/n1 experimentally and the perpendicularity of reflected and refracted rays

1823

— Fresnel derives the TIR phase shifts and folds them into the rhomb, converting linear to circular light with bare glass

1841

— Colladon guides light inside a falling water jet in Geneva; Tyndall's 1854 London demonstration makes it famous

1947

— Goos and Hänchen measure the lateral shift of a totally reflected beam — the evanescent skin made visible

1961

— Harrick and Fahrenfort independently launch ATR spectroscopy on the evanescent field

1966

— Kao and Hockham propose low-loss glass fiber; TIR becomes the backbone of telecommunications (Nobel 2009)

The physical picture

Start with the Brewster zero, because it has the best explanation in this chapter — sketched in one line when the oblique-incidence article first met the null, and made load-bearing here. What actually builds the reflected wave? The transmitted field drives the molecules of medium 2 into oscillation, and those oscillating dipoles re-radiate; the coherent sum of their back-radiation is the reflected beam (the same bookkeeping that made the transmitted beam travel at $c/n$ in the linear-media article). Now the geometry: a $p$ -polarized transmitted wave makes its dipoles oscillate in the plane of incidence, perpendicular to the refracted ray. An oscillating dipole radiates nothing along its own axis — its radiation pattern has a hole there (the full pattern is E9 business, Electric Dipole Radiation, deferred by name; the hole is all we need). So if the direction the reflected ray wants to go happens to line up with the dipole axis, the reflected wave has no one to build it. That alignment happens exactly when reflected and refracted rays are perpendicular, $\theta_B + \theta_T = 90°$ — and feeding that to Snell's law gives $\tan\theta_B = n_2/n_1$ in one line. The $s$ -channel's dipoles oscillate perpendicular to the plane of incidence, broadside to every ray in the problem; they radiate happily in the reflected direction at every angle, which is why no $s$ -Brewster exists. Total internal reflection needs no microscopics at all — it is Snell's law hitting a wall. Going dense to rare, the refracted ray bends away from the normal, running ahead of the incident one; at $\theta_c = \arcsin(n_2/n_1)$ it lies flat along the surface, and beyond that Snell demands $\sin\theta_T > 1$ . The equation does not break; it goes complex. The transmitted wavevector keeps its (real) tangential component — phase matching is non-negotiable — and pays for it with an imaginary normal component: a field that waves along the surface and decays into medium 2, dead within a wavelength. No net energy crosses; everything reflects. The two angles even come ordered: since $\tan\theta < \sin\theta$ never holds for acute angles, $\theta_B^{\rm internal} = \arctan(n_2/n_1) < \arcsin(n_2/n_1) = \theta_c$ always — climbing in angle from the dense side you first pass the quiet Brewster zero, then hit the wall.

Figure

The two landmark angles for glass to air — the Brewster zero at 33.7 degrees, the total-reflection wall at 41.8 degrees, and the evanescent skin beyond it

One habit worth forming now: for any interface pair, compute the three angles that organize everything. There are two Brewster angles — one from each side — and they are complementary ( $56.3°$ and $33.7°$ for air/glass: swapping the indices inverts the tangent, and complementary angles have reciprocal tangents), while the critical angle exists only on the dense side. Those three numbers are the complete "map" of the interface, and the limiting-cases table collects them for the standard pairs.

The two landmark conditions

The Brewster condition, derived below by recalling — not re-deriving — the previous article's $p$ -coefficient:

\tan\theta_B = \frac{n_2}{n_1}, \qquad \theta_B + \theta_T = 90°, \qquad (1)

where: $\theta_B$ is the incidence angle of zero $p$ -reflection and the second form is its geometric signature (reflected ⊥ refracted). Air→glass: $\theta_B = \arctan 1.5 = 56.3°$ ; glass→air: $33.7°$ ; air→water: $53.1°$ . At $\theta_B$ an unpolarized beam reflects as perfectly $s$ -polarized light — the only place in dielectric optics where a polarization is removed exactly rather than attenuated. Note what the formula does not contain: wavelength. Through dispersion $n(\omega)$ the Brewster angle drifts slightly across the visible (about half a degree for glass), which is why Brewster windows are "achromatic enough" for broadband lasers. The critical angle:

\sin\theta_c = \frac{n_2}{n_1} \qquad (n_1 > n_2), \qquad (2)

where: the condition exists only from the dense side — from the rare side Snell always has a solution and some light always leaks through. Glass→air: $41.8°$ (the reason $45°$ prisms totally reflect); water→air: $48.8°$ (the diver's "Snell's window": the whole sky compressed into a $97.5°$ cone overhead, mirror-dark water beyond it); diamond→air: $24.4°$ — diamond's tiny $\theta_c$ is why brilliants are cut so shallow a cone of angles escapes, and most light entering the crown ricochets internally before exiting upward as fire. Beyond $\theta_c$ the transmitted wavevector's normal component turns imaginary,

k_{2z} = i\kappa, \qquad \kappa = \frac{\omega}{c}\sqrt{n_1^2\sin^2\theta_I - n_2^2}, \qquad \mathbf{E}_2 \propto e^{-\kappa z}\,e^{i(k_x x - \omega t)}, \qquad (3)

where: the tangential wavenumber $k_x = (\omega/c)\,n_1\sin\theta_I$ stays real (phase matching is unconditional), so the field in medium 2 is a surface wave: propagating along $x$ , decaying in $z$ with penetration depth $1/\kappa$ — computed honestly in Derivation B, and sub-wavelength: for glass→air at $45°$ with red light (633 nm), $1/\kappa = 285$ nm; at $60°$ , 122 nm; only just above $\theta_c$ (at $42.5°$ ) does it stretch to 614 nm. This skin is where ATR spectroscopy lives and across which frustrated TIR tunnels. The reflection coefficients above $\theta_c$ become pure phases,

Figure

Computed evanescent decay into air for glass at 633 nanometers — the field one micron out is essentially gone, and steeper incidence thins the skin

r_s = \frac{\cos\theta_I - i\sqrt{\sin^2\theta_I - \sin^2\theta_c}}{\cos\theta_I + i\sqrt{\sin^2\theta_I - \sin^2\theta_c}}, \qquad |r_s| = |r_p| = 1, \qquad (4)

where: numerator and denominator are complex conjugates — unit modulus by inspection, which is total reflection stated algebraically. The physics moved into the arguments: each channel acquires its own reflection phase, and the relative phase $\Delta = \delta_p - \delta_s$ (Derivation C) is a free, exactly unitary polarization tool — the Fresnel rhomb converts linear to circular light with it, using nothing but a glass block and geometry.

Derivation ABrewster's angle mechanism-first, plus the complementary-pair theorem

The goal is (1), derived by the route the neighboring articles deliberately left to this one. The oblique-incidence article already extracted $\tan\theta_B = n_2/n_1$ algebraically — vanishing numerator of $r_p$ , Snell substitution, the $\sin^2\theta_B = n_2^2/(n_1^2+n_2^2)$ form — and that result is recalled by name, not repeated. Here the logic runs in the opposite, physical direction: start from the mechanism (dipoles cannot radiate along their axes), convert it to a geometric demand, and let Snell's law hand over the tangent law — then prove the complementary-pair theorem that the algebraic route leaves implicit. Two routes, one law; this is the direction you can reconstruct on a napkin years from now.

Step 1. The mechanism as a postulate. The reflected wave has no independent existence: it is the coherent back-radiation of medium 2's dipoles, driven by the transmitted field (the linear-media article's bookkeeping). For $p$ -polarization those dipoles oscillate in the plane of incidence, perpendicular to the refracted ray. A dipole radiates nothing along its own axis. Therefore the reflected wave must vanish whenever the reflected-ray direction coincides with the dipole axis — i.e. whenever the reflected ray is perpendicular to the refracted ray.

Step 2. Geometry to angles. Reflected ray at $\theta_B$ above the surface normal on one side, refracted ray at $\theta_T$ below on the other; perpendicularity of the two rays reads $\theta_B + \theta_T = 90°$ (draw it once: the angle between them is $180° - \theta_B - \theta_T$ ).

Step 3. Snell closes it. With $\theta_T = 90° - \theta_B$ , Snell's law becomes $n_1\sin\theta_B = n_2\sin(90° - \theta_B) = n_2\cos\theta_B$ , so

\tan\theta_B = \frac{n_2}{n_1}

— one line, no squaring, no radicals. Consistency audit against the algebraic route: the oblique-incidence article's $\sin^2\theta_B = n_2^2/(n_1^2 + n_2^2)$ is exactly $\tan^2\theta_B = n_2^2/n_1^2$ rearranged ✓. Why the formula is index-ratio-only: boundary conditions compare the media pairwise; no absolute scale can enter — a sanity check on any remembered version.

Step 4. The complementary-pair theorem (owned here). Swap the roles of the media — light arriving from the dense side — and the tangent law gives $\theta_B' = \arctan(n_1/n_2)$ . Since $\arctan x + \arctan(1/x) = 90°$ , the two Brewster angles of any interface are exactly complementary: $\theta_B + \theta_B' = 90°$ (glass: $56.3° + 33.7°$ ). Geometrically inevitable: the external Brewster's refracted ray travels at $\theta_T = \theta_B'$ , and reversibility of rays means the internal Brewster geometry is the same picture run backwards — one perpendicularity condition serving both directions. Practical reading: a Brewster window is Brewster-matched at both faces automatically (the internal face sits at $\theta_B'$ by Snell), which is why a single tilt angle makes the whole plate lossless for $p$ — the fact worked example 1 spends.

Step 5. What the mechanism forbids and permits. The $s$ -dipoles oscillate normal to the plane of incidence, broadside to every ray in the problem: no alignment is geometrically possible, hence no $s$ -Brewster in nonmagnetic media, at any angle, for any index pair. The loophole is magnetic response: with $\mu_1 \neq \mu_2$ , magnetic dipoles enter the re-radiation budget and the null can migrate to the $s$ -channel (the impedance-crossing language of the Fresnel article makes the same statement; the FAQ carries the metamaterial cash-out). And the polarizing payoff at the null: air→glass at $\theta_B$ has $R_s = 0.148$ — nothing of $p$ , 15% of $s$ per surface — the asymmetry that worked example 1 iterates into a polarizer. Verified: tangent law and perpendicularity reproduce Griffiths Eq. 9.118/Hecht §4.6 and the oblique-incidence article's algebraic result (routes agree exactly); complementarity checked numerically ( $\arctan 1.5 = 56.31°$ , $\arctan(1/1.5) = 33.69°$ , sum $90.00°$ ); dipole argument per Hecht §8.6.1; $R_s(\theta_B) = \sin^2(\theta_B - \theta_T)/\sin^2(\theta_B + \theta_T) = \sin^2(2\theta_B - 90°)$ evaluated: $\sin^2(22.6°) = 0.148$ ✓.

Go deeper · Derivation B · 2 minThe evanescent wave — decay length, zero net power, and total reflection

Derivation BThe evanescent wave — decay length, zero net power, and total reflection

The goal is the full above-critical package: the imaginary normal wavevector (3), the penetration depth with real numbers, the proof that no time-averaged power crosses the interface, and hence $|r| = 1$ in both channels. The strategy is phase matching plus the dispersion relation — no Fresnel algebra needed for the energy verdict, which is the elegant part. Watch how the instantaneous energy flow is not zero: the evanescent field is real and breathing; only its cycle average across the boundary vanishes.

Step 1. Phase matching. The boundary conditions must hold at every point of the interface for all time, so all three waves share $\omega$ and the tangential wavenumber: $k_x = (\omega/c)\,n_1\sin\theta_I$ — the kinematic package of the oblique-incidence article, recalled.

Step 2. Dispersion in medium 2. The transmitted field must satisfy medium 2's wave equation, so $k_x^2 + k_{2z}^2 = (n_2\,\omega/c)^2$ , giving

k_{2z}^2 = \left(\frac{\omega}{c}\right)^2\left(n_2^2 - n_1^2\sin^2\theta_I\right).

For $\theta_I > \theta_c$ the bracket is negative: $k_{2z} = i\kappa$ with $\kappa = (\omega/c)\sqrt{n_1^2\sin^2\theta_I - n_2^2}$ real. Why we keep only $+i\kappa$ : the root $-i\kappa$ grows exponentially with depth — inadmissible in a half-space with no sources at infinity. The math offered two solutions; physics vetoed one.

Step 3. The field and its skin. $\mathbf{E}_2 \propto e^{-\kappa z}e^{i(k_x x - \omega t)}$ : crests glide along the surface with phase velocity $\omega/k_x = c/(n_1\sin\theta_I) < c/n_2$ — slower than any propagating wave in medium 2 is allowed to be, which is another way to see why it cannot detach from the surface. Numbers for glass→air, $\lambda_0 = 633$ nm: $1/\kappa = 614$ nm at $42.5°$ , 285 nm at $45°$ , 122 nm at $60°$ — the skin thins rapidly beyond $\theta_c$ , and one micron into the air the field at $45°$ is down to 3% of its surface value.

Step 4. The energy verdict. The normal energy flux is $\langle S_z\rangle = \tfrac{1}{2}\,\mathrm{Re}[\mathbf{E}\times\mathbf{H}^*]_z$ . Every factor of the transmitted fields carries the same $e^{i k_{2z}z}$ , and the $z$ -flux assembles a factor $\mathrm{Re}(k_{2z}^{\,*})$ from the curl that builds $\mathbf{H}$ out of $\mathbf{E}$ — and $\mathrm{Re}(i\kappa)^* = \mathrm{Re}(-i\kappa) = 0$ :

\langle S_z\rangle_{\text{medium 2}} = 0.

Why this does not make the evanescent field fictitious: the instantaneous $S_z$ oscillates at $2\omega$ — energy sloshes into the skin and back out twice per cycle, netting zero. The tangential flux $\langle S_x\rangle$ is NOT zero: energy genuinely streams along the surface inside the skin, which is what the Goos–Hänchen shift (the reflected beam re-emerging slightly displaced along the surface) makes visible.

Step 5. Total reflection. Steady state, lossless media, no net flux into medium 2: energy conservation forces the reflected flux to equal the incident flux, $R = 1$ , for each polarization separately (the channels don't mix — mirror symmetry, previous article's FAQ). The algebraic confirmation is (4): above $\theta_c$ , Snell's law gives $\cos\theta_T = \tfrac{i}{\sin\theta_c}\sqrt{\sin^2\theta_I - \sin^2\theta_c}$ , and substituting this purely imaginary cosine into each Fresnel coefficient makes numerator and denominator complex conjugates, so $|r| = 1$ identically — with the leftover phase as Derivation C's subject. Verified: matches Jackson §7.4 and Zangwill §17.3; penetration depths recomputed from $\kappa = (2\pi/\lambda_0)\sqrt{n_1^2\sin^2\theta_I - n_2^2}$ : $\sqrt{2.25\times0.5 - 1} = 0.354$ → $1/\kappa = 633/(2\pi\times0.354) = 285$ nm at 45° ✓; the conjugate-pair structure of (4) checked by direct substitution; $\langle S_z\rangle = 0$ consistent with the Poynting machinery of this chapter's energy articles.

Go deeper · Derivation C · 2 minTIR phase shifts and the Fresnel rhomb

Derivation CTIR phase shifts and the Fresnel rhomb

The goal is the reflection phases hiding in (4) — the one part of total internal reflection that is neither "total" nor obvious — and their flagship application: Fresnel's rhomb, which manufactures circular polarization from a bare glass block. The strategy: write each coefficient as a conjugate pair, read off its half-angle tangent, form the relative phase between channels, and then design with it. This is also the honest foundation for the "phase tricks" of integrated photonics, where TIR phases set the modes of every waveguide.

Step 1. Phase of a conjugate pair. Any ratio $r = (a - ib)/(a + ib)$ with $a, b$ real has $|r| = 1$ and $\arg r = -2\arctan(b/a)$ . For the $s$ -channel, (4) gives $a = \cos\theta_I$ , $b = \sqrt{\sin^2\theta_I - \sin^2\theta_c}$ , so the $s$ phase shift is

\delta_s = -2\arctan\!\left(\frac{\sqrt{\sin^2\theta_I - \sin^2\theta_c}}{\cos\theta_I}\right).

The $p$ -channel's coefficient carries the extra index asymmetry, landing the same structure with $b$ enhanced by $1/\sin^2\theta_c$ :

\delta_p = -2\arctan\!\left(\frac{\sqrt{\sin^2\theta_I - \sin^2\theta_c}}{\sin^2\theta_c\,\cos\theta_I}\right).

Step 2. The relative retardation. What polarization optics uses is $\Delta = \delta_p - \delta_s$ ; a compact closed form follows from the arctan subtraction identity:

\tan\frac{\Delta}{2} = \frac{\cos\theta_I\,\sqrt{\sin^2\theta_I - \sin^2\theta_c}}{\sin^2\theta_I}.

Why $\Delta$ has a maximum: it vanishes at both ends — at $\theta_c$ (both channels reflect in phase, $b \to 0$ ) and at grazing (both saturate at $\pi$ ) — so somewhere between, the retardation peaks. For glass $n_1 = 1.51$ the peak is $45.9°$ at $\theta_I = 51.5°$ : a single internal reflection can retard by at most about an eighth-wave. Nature caps the per-bounce budget just above $45°$ — an accident of $n = 1.5$ glass that Fresnel spent brilliantly.

Step 3. Design the rhomb. Want a quarter-wave retarder ( $\Delta_{\rm total} = 90°$ )? One bounce cannot do it, but two bounces at $\Delta = 45°$ each can. Setting $\tan(\Delta/2) = \tan 22.5°$ for $n_1 = 1.51$ gives two solutions, $\theta_I = 48.6°$ and $54.6°$ (both above $\theta_c = 41.5°$ ✓); Fresnel chose $54.6°$ , where the curve is flatter and the tolerance kinder. A parallelogram prism with $54.6°$ face angles sends the beam through exactly two such bounces: linear light in at $45°$ to the plane of incidence emerges circular.

Step 4. Why bother, when wave plates exist. A birefringent quarter-wave plate retards by a material path difference $\propto \Delta n\,d/\lambda$ — right at exactly one wavelength. The rhomb's retardation depends only on $n(\lambda)$ through the weak dispersion of glass: across the visible it drifts by only a couple of degrees. It is the achromatic retarder — still the reference solution in broadband and ultrafast polarization optics, two centuries after Fresnel folded one out of a glass block to prove light's transversality. Verified: phase formulas match Born & Wolf §1.5.4 and Hecht §4.7; numerics recomputed — for $n = 1.51$ : $\Delta(48.6°) = 44.9° \approx 45°$ , $\Delta(54.6°) = 45.1° \approx 45°$ , maximum $45.9°$ at $51.5°$ ; for $n = 1.5$ the maximum is $45.2°$ , confirming how tight Fresnel's margin was; $\Delta \to 0$ at $\theta_c$ and at $90°$ ✓.

Limiting cases

Case	Result	Meaning
air→glass ( $n$ 1→1.5)	$\theta_B = 56.3°$ ; no $\theta_c$	Brewster only — TIR needs the dense side
glass→air	$\theta_B = 33.7°$ , $\theta_c = 41.8°$	zero first, wall second — always in that order
air→water ( $n = 1.33$ )	$\theta_B = 53.1°$	polarized glare geometry off lakes and roads
water→air	$\theta_c = 48.8°$	Snell's window: sky in a 97.5° cone above the diver
diamond→air ( $n = 2.42$ )	$\theta_c = 24.4°$	almost everything totally reflects: the cut's fire
$n_2 \to n_1$	$\theta_B \to 45°$ , $\theta_c \to 90°$	vanishing contrast: zero and wall retreat together
$\theta_I \to \theta_c^+$	$1/\kappa \to \infty$	evanescent skin thickens without bound at the edge
$\theta_I \to 90°$	$1/\kappa \to \lambda_0/(2\pi\sqrt{n_1^2 - n_2^2})$	thinnest possible skin — 96 nm for glass, red light
$\mu_1 \neq \mu_2$	$s$ -Brewster possible	impedance crossing can move channels — FAQ

Common trapTotal internal reflection does not mean zero field outside — and Brewster suppresses only one channel

Two misreadings, both expensive. First: "total" refers to the energy budget, not the field. A real, measurable evanescent field extends a few hundred nanometers into the forbidden medium; anything brought into that skin — a fingerprint ridge, a protein layer, a second prism — couples to it and breaks the totality. Fingerprint scanners, ATR spectrometers, and evanescent biosensors are commercial proof that the field is there; a TIR design that lets contaminants (water: $\theta_c$ jumps from $41.8°$ to $62.5°$ !) or mounting adhesive touch the reflecting face will leak catastrophically. Second: Brewster's angle zeroes $r_p$ only — an unpolarized beam still reflects ( $R_s = 0.148$ for glass), just purely $s$ -polarized. "No reflection at Brewster" is true only for light already $p$ -polarized. And the two angles answer different questions in different directions: Brewster exists from both sides of the interface (complementary values), the critical angle only from the dense side. Keep the interface's three-angle map straight before designing anything.

Worked examples

1 · The pile-of-plates polarizer, quantified (confidence builder)

The oblique-incidence article told the Brewster-window story qualitatively (tilted laser windows, lossless for $p$ , output polarized). Make it quantitative, then stack it: for glass plates ( $n = 1.5$ ) at Brewster incidence, compute the per-surface $s$ -loss, and find how many plates a pile-of-plates polarizer needs for a given purity.

Step 1. The per-surface $s$ -reflectance at $\theta_B = 56.3°$ , from the trig form at the Brewster condition ( $\theta_B + \theta_T = 90°$ makes the denominator $\sin(\theta_B + \theta_T) = 1$ ): $R_s = \sin^2(\theta_B - \theta_T) = \sin^2(22.6°) = \mathbf{0.148}$ . The $p$ -channel: exactly zero, at both faces of the plate — the complementary-pair theorem of Derivation A guarantees the internal face (hit at $33.7°$ ) is also at Brewster.

Step 2. One plate = two surfaces: $s$ -transmission $0.852^2 = 0.726$ per plate, $p$ -transmission $1.000$ . The 26% per-plate tax on $s$ is what suppresses the $s$ -mode in a laser cavity (round-trip gain in a HeNe is a few percent — no contest); as a polarizer for a single pass, though, one plate is feeble: the transmitted beam is only mildly polarized.

Step 3. Stack five plates (ten surfaces): $T_s = 0.852^{10} = \mathbf{0.20}$ , $T_p = 1.00$ . Transmitted degree of polarization: $(T_p - T_s)/(T_p + T_s) = 0.80/1.20 = \mathbf{0.67}$ .

Step 4. Scale to purity: each plate multiplies the $s$ -leak by 0.726, so DOP $= (1 - 0.726^N)/(1 + 0.726^N)$ ; reaching 99% ( $0.726^N = 0.005$ ) takes $N = \ln 0.005/\ln 0.726 \approx \mathbf{17}$ plates... more precisely 16.6, so 17. A seventeen-plate stack is bulky and alignment-hungry — which is why pile-of-plates polarizers survive only where absorptive and crystal polarizers fail (high-power infrared lasers, vacuum-UV), while the Brewster window — which needs the $p$ -channel perfect rather than the $s$ -channel dead — is everywhere. The design lesson generalizes: Brewster geometry gives an exact zero in one channel but only a modest tax on the other; exact extinction and strong rejection are different engineering problems.

2 · The 45° prism: dry, wet, and how deep the "forbidden" field reaches (exam level)

Binocular prisms turn beams by TIR at $45°$ internal incidence ( $n_{\rm glass} = 1.5$ , red light $\lambda_0 = 633$ nm). (a) Verify TIR works, and find the penetration depth into the air. (b) The prism's face gets wet ( $n_{\rm water} = 1.33$ ). What happens? (c) A second prism approaches the first across a 100 nm air gap — estimate the transmitted fraction.

Step 1. Dry check: $\theta_c = \arcsin(1/1.5) = 41.8° < 45°$ ✓ — total reflection with no metallic coating and no absorption loss, which is exactly why prisms outperform mirrors inside optical instruments.

Step 2. Penetration depth: $\kappa = \tfrac{2\pi}{\lambda_0}\sqrt{n_1^2\sin^2 45° - 1} = \tfrac{2\pi}{633\,\text{nm}}\sqrt{1.125 - 1} = 3.51\times10^6\ \text{m}^{-1}$ , so $1/\kappa = \mathbf{285\ nm}$ — the field skims less than half a wavelength into the air before dying. One micron out, it is down to $e^{-3.5} \approx 3\%$ .

Step 3. Wet face: the outer medium becomes water, $\theta_c' = \arcsin(1.33/1.5) = \mathbf{62.5°}$ . Now $45° < \theta_c'$ — TIR is destroyed outright. The interface reverts to ordinary Fresnel splitting (a few percent reflected), and the prism dumps most of the beam into the water film. A drop of water, a fingerprint, or index-matched cement on a TIR face is not a small perturbation; it is an off switch. This is also the sensing principle run backward: a fingerprint scanner images exactly the ridge-contact points where TIR died.

Step 4. Frustrated TIR across the gap: with the second glass surface a distance $d$ inside the evanescent skin, the transmitted power scales as $e^{-2\kappa d}$ (amplitude reaches the far side as $e^{-\kappa d}$ , power squares it — the exact coupled-boundary calculation adds prefactors of order one). For $d = 100$ nm: $e^{-2\times3.51\times10^6\times10^{-7}} = e^{-0.70} \approx \mathbf{0.50}$ — half the light tunnels across a gap it cannot classically cross. At $d = 500$ nm the fraction is 3%; by two microns it is gone. Beam-splitter cubes with tunable air gaps, optical touchscreens, and the standard classroom analogy for quantum tunneling (the mathematics is the same exponential) all live on this curve.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §9.3.3 (Reflection and Transmission at Oblique Incidence) · Jackson, Classical Electrodynamics, 3rd ed., §7.3–7.4 · Hecht, Optics, 5th ed., §4.6 (Fresnel Equations) and §4.7 (Total Internal Reflection) · Zangwill, Modern Electrodynamics, §17.3 · Born & Wolf, Principles of Optics, 7th ed., §1.5

Provenance. Brewster condition $\tan\theta_B=n_2/n_1$ and the $\theta_B+\theta_t=90^\circ$ geometry verified against Griffiths §9.3.3 and Hecht §4.6; critical angle $\sin\theta_c=n_2/n_1$ and the evanescent/zero-flux analysis against Jackson §7.4 and Zangwill §17.3. Numerical values (air-glass $56.3^\circ$ , glass-air $\theta_c=41.8^\circ$ , $\theta_B=33.7^\circ$ , water $\theta_c=48.8^\circ$ ) are standard textbook results; a verifier should recompute $\arctan 1.5$ , $\arcsin(1/1.5)$ , $\arctan(1/1.5)$ , $\arcsin(1/1.33)$ and the Brewster derivation algebra. Fresnel rhomb retardation per reflection (~45° near 54° in $n\approx1.5$ glass) is from Born & Wolf §1.5. Addendum for this edition: scope split executed against BOTH landed neighbors — the Fresnel coefficients are recalled by name (component route: the oblique-incidence article; impedance route: The Fresnel Equations), the algebraic Brewster extraction ( $\sin^2\theta_B$ form) is the oblique-incidence article's and is cited not repeated; this article derives the law mechanism-first (dipole postulate → perpendicularity → tangent law), owns the complementary-pair theorem $\theta_B + \theta_B' = 90°$ , and owns the full TIR package including the evanescent wave and the reflection phases. Pile-of-plates recomputed honestly: DOP after N plates $= (1-0.726^N)/(1+0.726^N)$ — five plates 0.67, 99% purity needs 17 plates; $R_s(\theta_B) = \sin^2(2\theta_B - 90°) = 0.148$ for glass. Worked numbers recomputed from scratch: penetration depths at $\lambda_0 = 633$ nm glass→air — $1/\kappa = 614$ nm ( $42.5°$ ), 285 nm ( $45°$ ), 122 nm ( $60°$ ), 96 nm limit at grazing; frustrated-TIR transmission $e^{-2\kappa d} = 0.70/0.50/0.25/0.03$ at gaps 50/100/200/500 nm; rhomb solutions for $n = 1.51$ : $\Delta = 45°$ at $\theta_I = 48.6°$ and $54.6°$ , maximum $45.9°$ at $51.5°$ ; pile-of-plates $T_s = 0.852^{10} = 0.20$ , transmitted DOP 0.67; wet-prism critical angle $\arcsin(1.33/1.5) = 62.5°$ ; diamond $\theta_c = \arcsin(1/2.42) = 24.4°$ .

Builds on: The Fresnel Equations · Reflection and Transmission at Oblique Incidence · Polarization of Electromagnetic Waves · Electromagnetic Waves in Linear Dielectric Media · Complex Notation for Electromagnetic Waves

Leads to: Jones Vectors and Stokes Parameters · Energy Density and the Poynting Vector · Intensity and Time-Averaged Quantities

What does this article own versus the two reflection articles before it?

The trio divides so that nothing is derived twice. The oblique-incidence article owns the kinematics and the first pass through the dynamics: both Fresnel coefficients in the standard component frame, with the Brewster angle extracted algebraically from $r_p$ 's vanishing numerator and TIR stated with a forward pointer. The Fresnel Equations owns the coefficient package in depth: the unified impedance route (including magnetic media), the trigonometric forms, the energy-ledger theorem, conventions, and the signed-amplitude anatomy. This article owns the two landmark phenomena end to end: Brewster by the mechanism-first route (dipole postulate → geometry → tangent law, plus the complementary-pair theorem) with its polarizing applications quantified, and the entire above-critical regime — evanescent wave, penetration depths, zero-net-flux proof, TIR phases, the rhomb, frustrated TIR. When this article needs a coefficient, it recalls it by name and spends it. Where do the coefficients come from? Two articles back (component route) or one (impedance route); the algebra of the Brewster null lives back there too, and its physics lives here.

Is the evanescent wave actually real, or just a mathematical artifact of a complex angle?

Emphatically real — it is one of the most-instrumented fields in physics. Direct evidence: frustrated TIR (light tunnels across sub-micron gaps, with the measured exponential $e^{-2\kappa d}$ of worked example 2), ATR spectroscopy (molecules parked in the skin absorb from it, and their spectra appear in the reflected beam), TIRF microscopy (fluorophores light up only inside the ~100 nm slice), and the Goos–Hänchen shift (the reflected beam exits displaced along the surface by roughly the skin depth, measured in 1947 by trapping a beam in many successive TIRs). What is not real is net energy transport across the boundary: the normal Poynting flux averages to zero, with energy sloshing in and out twice per optical cycle. "Zero average flux" and "zero field" are different statements, and the entire evanescent-sensing industry lives in the gap between them.

Why does the Brewster zero exist only for p-polarization — and can any trick create an s-Brewster?

In nonmagnetic media, never: the $s$ -channel's effective impedance mismatch ( $Z/\cos\theta$ form, previous article) is monotonic in angle, and microscopically the $s$ -dipoles radiate broadside to the reflected direction at every angle, so nothing can silence them. The loophole is magnetic response. With $\mu_1 \neq \mu_2$ the impedance-crossing condition can be met in the $s$ -channel instead (or in both, or neither): set the general $Z$ -form coefficients of the previous article to zero and an $s$ -Brewster condition emerges involving both $\varepsilon$ and $\mu$ ratios. Natural materials at optical frequencies have $\mu \approx \mu_0$ , so this stayed a curiosity for a century — until metamaterials made $\mu(\omega)$ an engineering variable; "generalized Brewster angle" surfaces that kill chosen polarizations at chosen angles are now a standard metasurface design product. The rule of thumb survives: a reflection zero is an impedance crossing, and you can move it to whichever channel you can engineer.

The penetration depth diverges as the angle approaches the critical angle — does the reflection stop being total there?

Reflection is exactly total for every $\theta_I > \theta_c$ in the ideal problem (infinite plane wave, infinitely wide interface, lossless media): $|r| = 1$ with no exceptions, even though $1/\kappa \to \infty$ at the edge. What the diverging skin does change is robustness and finite-size behavior. A real beam has finite width and angular spread; components straddling $\theta_c$ leak, and just above $\theta_c$ the enormous skin makes the configuration exquisitely sensitive to anything within reach — absorbing media, curvature, a second interface meters of skin away. That sensitivity is exploited (sensors park just above $\theta_c$ for maximum response) and suffered (fiber bends leak precisely because curvature tilts local incidence angles back toward $\theta_c$ while the evanescent tail reaches into the jacket). Design rule: deep TIR ( $\theta_I \gg \theta_c$ , thin skin) for robust mirrors; edge TIR (thick skin) for sensing.

Why is the internal Brewster angle always below the critical angle?

Because $\tan\theta > \sin\theta$ for every acute angle, so the angle whose tangent equals $n_2/n_1$ is smaller than the angle whose sine does: $\arctan(n_2/n_1) < \arcsin(n_2/n_1)$ , always, for any index pair. The physical consequence: climbing in internal angle you always meet the quiet zero before the wall, and the two phenomena never compete for the same geometry. It also means a Brewster-cut internal face (33.7° for glass) can never be a TIR face, and vice versa — useful when debugging why a "Brewster window" inside a prism assembly is leaking (it must leak: it transmits by design) or why a TIR face shows no polarization selectivity (above $\theta_c$ both channels reflect completely; only their phases differ).

Snell's window: what exactly does a diver see at and beyond 48.8 degrees?

Looking up, every ray arriving from the whole sky above refracts into a cone of half-angle $\theta_c = 48.8°$ around the vertical — the hemisphere of sky compressed into "Snell's window," with the horizon mapped to the cone's rim (and strongly darkened and color-fringed there, since transmission plummets and dispersion spreads $\theta_c$ slightly across colors). Beyond the rim, geometry forbids any ray from above: those directions satisfy the TIR condition in reverse, so the surface acts as a mirror and the diver sees the reflected underwater scene — the bottom, fish, her own bubbles. The window is textbook Fresnel physics made panoramic, and underwater photographers compose with it deliberately. Waves wrinkle the surface and shatter the rim into the dancing caustics on every pool floor — each bright filament is the window's edge, refocused.

Is frustrated TIR really the same mathematics as quantum tunneling?

Structurally, yes — remarkably closely. The evanescent region is governed by $d^2E/dz^2 = \kappa^2 E$ , formally identical to the Schrödinger equation under a barrier with $\kappa = \sqrt{2m(V-E)}/\hbar$ ; the transmitted fraction decays as $e^{-2\kappa d}$ in both problems; and the transmitted wave emerges with the incident frequency/energy intact, having "borrowed" passage through a classically forbidden zone. The analogy runs deep enough that optics experiments are used to study tunneling questions (traversal-time debates were partly fought with microwave frustrated-TIR setups). The honest differences: the optical problem is a classical field with no probabilistic interpretation — the "tunneling fraction" is a literal power fraction, not a chance per photon until you quantize; and the photon picture adds no new prediction here since Maxwell already is the exact "wave mechanics" of light. As an intuition pump in either direction, it is the best analogy in physics that is also a working device: your phone's proximity sensor may be running it right now.