# Snell's law **Snell's law** states that when [[Electromagnetic_radiation|light]] crosses a boundary between two transparent media, the angles the ray makes with the normal satisfy n₁·sin(θ₁) = n₂·sin(θ₂), where n is the [[Refractive_index|refractive index]] of each medium. It is the whole of [[Refraction|refraction]] in one line: the sine, not the angle, is what scales. In the microsim below the reader drags the angle of incidence from 0° to 90° and the index ratio n₂/n₁ from 0.5 to 2.5, and a fan of two hundred rays shows the consequences — the ray bending toward the normal on the way into glass, away from it on the way out, [[Total_internal_reflection|total internal reflection]] switching on abruptly past sin(θ_c) = n₂/n₁, and a raindrop preset that stacks refraction, one internal reflection and refraction again to produce the [[Rainbow|rainbow]] at the minimum of D(θ) = 180° + 2·θ − 4·asin(sin(θ)/n). The law is known in Europe for Willebrord Snellius, who found it around 1621 and did not publish, but the earliest surviving statement is by [[Ibn_Sahl_(mathematician)|Ibn Sahl]] in Baghdad in 984, six centuries before. On the [[Physics|Physics]] flagship this article serves Part I — History at *Medieval European and Islamic* (row P4), where it is the decisive case: a quantitative law, correct in modern form, discovered by measuring a hypotenuse ratio in a drawing of a lens, and then lost and refound three times. ## History Ptolemy's *Optics*, in the second century AD, contains the first tabulated refraction data — angle pairs for air-to-water, air-to-glass and water-to-glass at ten-degree steps — fitted by a rule in which the refracted angle is a quadratic function of the incident angle.[^ptolemy] The rule is a display fit, ILLUSTRATIVE rather than physical: it reproduces the table over the measured range and fails outside it, and there is reason to think the table itself was smoothed to match. Around 984, in a Baghdad manuscript *On Burning Instruments*, Ibn Sahl drew a ray crossing a plane face of a lens and marked the two hypotenuses of the right triangles built on the incident and refracted segments. He treated the ratio of those hypotenuses as a constant of the material, and used it to design a hyperbolic lens that focuses without aberration. The ratio of hypotenuses is exactly the ratio of the sines, so the statement is Snell's law in geometric dress, applied immediately to an engineering problem.[^rashed] [[Ibn_al-Haytham|Ibn al-Haytham]], working in Cairo a generation later, built the great synthesis of the *Book of Optics* on the principle that vision proceeds from object to eye and on systematic experiment with apertures and refracting vessels; he did not, however, recover Ibn Sahl's constant ratio.[^alhazen] The rainbow came apart in two places at once around 1300. [[Kamāl_al-Dīn_al-Fārisī|Kamāl al-Dīn al-Fārisī]], commenting on Ibn al-Haytham, and Theodoric of Freiberg in Germany, independently modelled a raindrop by a large water-filled glass sphere, traced refraction into it, reflection at the far surface and refraction out, and so explained both the primary bow and the reversed colours of the secondary bow from two internal reflections.[^boyer] The same Latin schools that received this work through translation produced Jean Buridan's impetus, and the pairing is not accidental: both are attempts to replace a qualitative Aristotelian account with a rule that can be traced step by step.[^clagett] Thomas Harriot found the sine law in 1602 and told no one; Snellius rediscovered it around 1621 and left it in manuscript; [[Christiaan_Huygens|Huygens]] saw Snellius's papers and named it for him.[^sabra] The first publication is Descartes's *Dioptrique* of 1637, which derives the law from an analogy with a ball crossing a surface, and whose companion essay on meteors computes the rainbow's angle correctly.[^descartes] Fermat objected that Descartes's derivation implicitly makes light travel *faster* in the denser medium, and in 1662 obtained the same law from the principle of least time with light travelling *slower* there — the same formula from an opposite physical assumption.[^fermat] Huygens's *Traité de la lumière* of 1690 derived it a third way, from secondary wavelets.[^huygens] ## Explanation A ray that crosses from air into water at 45° does not simply bend by some fixed amount. What is conserved is n·sin(θ), so with n₁ = 1.00 and n₂ = 1.333 the transmitted angle is asin(sin 45°/1.333) = 32.0°, and with crown glass at n₂ = 1.52 it is 27.7° (derived).[^b079-ch1] Push the incidence toward 90° and the transmitted angle does not reach 90°: it saturates at asin(n₁/n₂), which is 48.6° for water and 41.1° for crown glass, so all of the outside world arrives inside a cone of that half-angle — the fisheye window a swimmer sees looking up. The microsim runs this directly. Its HUD carries `n1*sin(theta1) = n2*sin(theta2)`, and it has two controls: the incidence angle θ₁ from 0° to 90°, and the index ratio n₂/n₁ from 0.5 to 2.5 with presets for water (1.333), crown glass (1.52) and diamond (2.417).[^b079-ch1] A fan of two hundred rays leaves a point source, so the reader sees the whole family bend at once rather than one representative ray, and the fraction of the fan that survives into the second medium shrinks visibly as the ratio drops below 1. Three readouts follow the slider: θ₂, the critical angle when n₂/n₁ < 1, and the lateral displacement of a ray through a parallel-sided slab, d = t·sin(θ₁ − θ₂)/cos(θ₂), which is why a spoon in a glass of water looks broken rather than bent. ## Derivations and formula The law is over-determined in a useful way: it follows from a variational principle, from a wave construction, from the boundary conditions of [[Maxwell's_equations|electromagnetism]] and from a conservation argument, and each derivation says something the others do not. ### Derivation from Fermat's principle Put the source at (0, a) in medium 1 and the target at (d, −b) in medium 2, and let the ray cross the boundary at (x, 0). The travel time is T(x) = n₁·√(x² + a²)/c + n₂·√((d − x)² + b²)/c. Setting dT/dx = 0 gives n₁·x/√(x² + a²) = n₂·(d − x)/√((d − x)² + b²), and each fraction is the sine of the corresponding angle to the normal, so n₁·sin(θ₁) = n₂·sin(θ₂).[^fermat] The derivation explains why the sine appears — it is the derivative of a path length with respect to the crossing point — and it fixes the direction of the effect: light must travel slower in the medium with the larger index, c/n. That was the point Fermat won against Descartes, and it was not confirmed by measurement until Léon Foucault compared the [[Speed_of_light|speed of light]] in air and in water in 1850 and found it lower in water.[^foucault] As a piece of the [[Calculus_of_variations|calculus of variations]], it is the optical ancestor of the action principles of mechanics. ### Derivation from Huygens's principle Treat the incoming [[Plane_wave|plane wave]] as a front whose every point is a source of secondary wavelets. While the edge of the front still in medium 1 travels a distance v₁·Δt along the surface trace, the edge already in medium 2 travels v₂·Δt. Both distances share the same hypotenuse L along the interface, so sin(θ₁) = v₁·Δt/L and sin(θ₂) = v₂·Δt/L, and dividing gives sin(θ₁)/sin(θ₂) = v₁/v₂ = n₂/n₁.[^huygens] The wave derivation makes plain what the ray picture hides: refraction is what a [[Wave|wave]] does when one end of its front is slowed before the other, and the same construction with v₂ = v₁ gives the law of reflection. ### Derivation from Maxwell's equations At a plane interface the tangential components of the electric and magnetic fields must be continuous everywhere on the surface and at all times. That is possible only if the incident, reflected and transmitted waves share the same frequency and the same tangential wavevector component, k₁·sin(θ₁) = k₂·sin(θ₂). Since k = n·ω/c, this is Snell's law, and it arrives without any assumption about rays or least time.[^b079-ch1] The same boundary conditions, carried further, give the Fresnel coefficients for how much light is reflected and how much transmitted, and the polarisation dependence that Snell's law itself says nothing about. The phase-matching argument is the most general of the four: it holds for any wave at any interface, and it is why [[Photonic_crystal|photonic crystals]] and [[Negative-index_metamaterial|negative-index metamaterials]] can be described as having an effective index at all. ### Derivation from conservation of energy and momentum Read as [[Photon|photons]], the same statement is a conservation law. The boundary is uniform in time, so [[Energy|energy]] ħω is conserved and the frequency does not change on crossing. The boundary is uniform along its own plane, so the component of [[Momentum|momentum]] parallel to it is conserved: ħ·k₁·sin(θ₁) = ħ·k₂·sin(θ₂). The perpendicular component is not conserved, because the medium is not uniform in that direction, and that missing conservation is the refraction. Stated this way the law is a corollary of translational symmetry rather than a property of light, which is why the acoustic and the matter-wave versions have the same form. ### Vector form For ray tracing the angular form is inconvenient. With d̂ the unit incident direction, n̂ the unit normal pointing back into medium 1, and r = n₁/n₂, define c₁ = −n̂·d̂ and c₂ = √(1 − r²·(1 − c₁²)). The transmitted direction is then t̂ = r·d̂ + (r·c₁ − c₂)·n̂, and the reflected direction is d̂ + 2·c₁·n̂. The quantity under the square root goes negative exactly when the incidence is beyond the critical angle, so one branch test handles both refraction and total internal reflection — which is how the microsim's two hundred rays are computed each frame, and how every renderer in [[Optical_engineering|optical engineering]] does it. ## Total internal reflection and critical angle When light travels toward a medium of lower index, the transmitted angle is larger than the incident angle and reaches 90° at the critical angle sin(θ_c) = n₂/n₁. Beyond it no real solution exists and the boundary becomes a perfect mirror — perfect in a sense no metal coating achieves, since nothing is absorbed. For water to air θ_c is 48.6°, for crown glass 41.1°, for [[Fibre-optic_gyroscope|fibre]]-grade fused silica about 44°, and for diamond 24.4°, which is why a brilliant cut traps light through several internal bounces before releasing it and why diamond looks the way it does (derived).[^b079-ch1] The 45°–45°–90° prism is the practical consequence: at 45° incidence, glass-to-air exceeds every one of those critical angles, so the prism turns a beam through 90° with no coating and no loss, and two of them turn it through 180°. Beyond θ_c the field does not simply stop at the surface. A non-propagating evanescent wave penetrates the second medium and decays over roughly a wavelength, which is why bringing a second prism within a fraction of a micrometre lets light cross the gap — frustrated total internal reflection, the optical analogue of quantum tunnelling. ## Dispersion The refractive index depends on wavelength, so Snell's law sends different colours in different directions and a single ray entering a prism leaves as a spectrum. For water the index runs from about 1.331 in the red to about 1.343 in the violet.[^b079-ch1] Over the visible range the variation is usually fitted by Cauchy's equation n(λ) = A + B/λ², which is an empirical display fit with no derivation behind it and is marked ILLUSTRATIVE where the sim uses it; the physical account requires the resonances of the medium and is given by the Sellmeier form. The microsim's raindrop preset makes the consequence visible. A ray entering a spherical drop at impact parameter set by θ, refracting, reflecting once at the back and refracting out again, leaves deviated by D(θ) = 180° + 2·θ − 4·asin(sin(θ)/n), a form that is standard geometric optics rather than a display taken from a Portal Book.[^b079-ch1] D has a minimum, so rays pile up at that deviation and the drop throws back a bright cone. Solving numerically at n = 1.333 gives θ = 59.4° and D_min = 137.9°, that is 42.1° from the antisolar point; with the red index 1.331 the bow sits at 42.4° and with the violet index 1.343 at 40.7°, so the primary bow is about 1.7° wide with red outside (all derived). The same calculation with two internal reflections, D₂(θ) = 360° + 2·θ − 6·asin(sin(θ)/n), gives 50.4° for red and 53.5° for violet: the secondary bow, fainter, wider and colour-reversed, with the unlit gap of Alexander's dark band between 42° and 50° where no minimum-deviation ray emerges (derived). This is al-Fārisī's and Theodoric's result, and the arithmetic that produces it is one line of Snell's law applied three times. ## Lossy, absorbing, or conducting media In an absorbing or [[Electrical_resistivity_and_conductivity|conducting]] medium the index is complex, ñ = n + i·κ, with κ the extinction coefficient that sets how fast the amplitude decays. Snell's law survives unchanged in form — ñ₁·sin(θ₁) = ñ₂·sin(θ₂) — but the transmitted angle is now a complex number and no longer names a direction. What the transmitted wave becomes is inhomogeneous: its planes of constant phase and its planes of constant amplitude are no longer parallel, the first tilted by refraction and the second lying parallel to the surface. The direction in which energy actually flows must then be computed from the time-averaged Poynting vector rather than read off from an angle. For a good conductor the imaginary part dominates, so light entering from air is refracted almost to the normal whatever the incidence, and the amplitude falls by a factor e over the skin depth δ = √(2/(μ·σ·ω)). The practical reading is that a metal surface has no useful transmitted ray at all, only a thin absorbing layer and a strong reflection, and the same complex formalism covers the evanescent field beyond the critical angle discussed above, where n is real but the cosine is imaginary. These are standard forms: the corresponding displayed equations in Portal Book 079 were lost in text extraction, and they should be verified against the printed pages before being quoted as the book's own.[^b079-ch1][^b079-geom] ## See also - [[Refraction]] - [[Rainbow]] - [[Total_internal_reflection]] - [[Refractive_index]] - [[Ibn_Sahl_(mathematician)]] - [[Ibn_al-Haytham]] - [[Christiaan_Huygens]] ## References [^b079-ch1]: Ling, Samuel J.; Sanny, Jeff; Moebs, William (2016). *University Physics Volume 3*. OpenStax. Chapter 1, "The Nature of Light", pp. 15–58 (the law of refraction, the table of refractive indices including water 1.333, crown glass 1.52 and diamond 2.419, total internal reflection and the critical angle, dispersion and the rainbow; page to pin). Portal Book 079. Nearly every displayed equation in this volume was lost in text extraction, so n₁·sin(θ₁) = n₂·sin(θ₂), sin(θ_c) = n₂/n₁, the slab displacement and the rainbow deviation are quoted here as standard forms, to be verified against the PDF pages. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^b079-geom]: Ling, Sanny and Moebs (2016), *University Physics Volume 3*, Chapter 2, "Geometric Optics and Image Formation", pp. 59–118 (ray tracing through refracting surfaces; page to pin). Portal Book 079; same extraction caveat as above. [^rashed]: Rashed, Roshdi (1990). "A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses." *Isis* 81 (3) (pages to pin). The reconstruction of Ibn Sahl's *Fī al-ālāt al-muḥriqa* (c. 984) and of the constant hypotenuse ratio equivalent to the sine law. [^ptolemy]: Ptolemy, Claudius. *Optics*, Book V (tables of refraction for air–water, air–glass and water–glass). English translation: Smith, A. Mark (1996). *Ptolemy's Theory of Visual Perception*. Philadelphia: American Philosophical Society (page to pin). [^alhazen]: Ibn al-Haytham. *Kitāb al-Manāẓir* (*Book of Optics*), c. 1011–1021. English translation: Smith, A. Mark (2001). *Alhacen's Theory of Visual Perception*. Philadelphia: American Philosophical Society (page to pin). [^boyer]: Boyer, Carl B. (1959). *The Rainbow: From Myth to Mathematics*. New York: Thomas Yoseloff (Kamāl al-Dīn al-Fārisī's and Theodoric of Freiberg's water-sphere models of the primary and secondary bows, c. 1300–1310; page to pin). [^clagett]: Clagett, Marshall (1959). *The Science of Mechanics in the Middle Ages*. Madison: University of Wisconsin Press (Buridan's impetus and the Latin reception of the Arabic scientific corpus; page to pin). [^descartes]: Descartes, René (1637). *La Dioptrique* and *Les Météores*, in *Discours de la méthode*. Leiden: Ian Maire. Discourse 2 of the *Dioptrique* (the law of refraction) and Discourse 8 of the *Météores* (the rainbow); page to pin. [^fermat]: Fermat, Pierre de (1662). "Analysis ad refractiones" and "Synthesis ad refractiones", in Tannery, Paul; Henry, Charles, eds. (1891–1912), *Œuvres de Fermat*, Paris: Gauthier-Villars (page to pin). [^huygens]: Huygens, Christiaan (1690). *Traité de la lumière*. Leiden: Pieter van der Aa. Chapter 3, on refraction (page to pin). [^sabra]: Sabra, A. I. (1981). *Theories of Light from Descartes to Newton*. Cambridge: Cambridge University Press (Harriot's unpublished 1602 determination, Snellius's manuscript of about 1621, Descartes's publication of 1637 and the Fermat–Descartes dispute; page to pin). [^foucault]: Foucault, Léon (1850). "Méthode générale pour mesurer la vitesse de la lumière dans l'air et les milieux transparents." *Comptes rendus hebdomadaires des séances de l'Académie des sciences* 30 (pages to pin). ## External links - [*University Physics Volume 3*](https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3), OpenStax — Portal Book 079, chapter 1 "The Nature of Light" and chapter 2 "Geometric Optics and Image Formation" - The Wikipedia pair's external links list further open sources on refraction and ray tracing; no other canonical open source is asserted here. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Snell's_law.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Snell's law* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Snell's_law.html" data-title="Snell's law"></div> *Built from `MICROSIM_GUIDE/specs/sims/Snell's_law.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Snell's_law) : [Wikitube](https://en.wikitube.io/wiki/Snell's_law) · pinned revision [1372013483](https://en.wikipedia.org/w/index.php?oldid=1372013483) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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