Snell's Law and Total Internal Reflection: How Optical Fibre Guides Light

Snell's law says that when light crosses a boundary between two transparent materials, n₁ sin θ₁ = n₂ sin θ₂, where n is each material's refractive index and θ is the angle from the normal. Going from a denser to a less dense material, the ray bends away from the normal, and beyond a critical angle no light gets out at all. That total internal reflection is what keeps light trapped inside the glass core of an optical fibre for tens of kilometres.

This post goes one level deeper than our explainer on Maxwell's equations, from Faraday to radio, Wi-Fi and fiber, which shows why light is an electromagnetic wave in the first place, and the electromagnetic spectrum explained, which places the infrared wavelengths that fibre uses. Where those posts stay broad, this one does the arithmetic: refraction angles for water and glass, critical angles for a table of materials, the numerical aperture of a real standardised fibre, and the reason single mode and multimode fibres look so different. If you want the manufacturing side, how fiber optic cables are made and laid and how optical transceivers are made pick up where this one stops.

A note on numbers. The speed of light comes from NIST's CODATA page. Indices for common materials come from the HyperPhysics optics pages at Georgia State University, fused silica from Malitson's 1965 dispersion formula, and water from the IAPWS release on its refractive index. Fibre parameters come from the current ITU-T Recommendations G.652 (08/2024), G.657 (08/2024) and G.651.1 (11/2018). Values we chose ourselves are labelled illustrative, and every calculation was run with the script in the walkthrough.

Key takeaways

• Refractive index is the speed of light in vacuum divided by its speed in the material, n = c/v. Water is about 1.333, fused silica about 1.458 in yellow light and 1.444 at 1550 nm, diamond 2.417.
• Snell's law, n₁ sin θ₁ = n₂ sin θ₂, fixes the bent angle. Light entering water at 30° from the normal continues at 22.0°; entering fused silica it continues at 20.1°.
• The critical angle is θc = arcsin(n₂/n₁), and only exists going from higher to lower index. For fused silica to air it is 43.3°; for water to air 48.6°; for diamond to air 24.4°.
• Total internal reflection returns all of the light beyond θc, though an evanescent field reaches a few hundred nanometres past the boundary, the physics behind fingerprint sensors and fibre taps.
• Numerical aperture NA = √(n₁² − n₂²) sets how wide a cone of light a fibre accepts. The ITU-T G.651.1 multimode fibre has a nominal NA of 0.20, an acceptance half-angle of 11.5°.
• Single mode fibre (ITU-T G.652) has a small core, around 9 µm of mode field diameter at 1310 nm in a 125 µm cladding, so only one mode travels and modal dispersion vanishes. Multimode fibre (G.651.1) uses a 50 µm graded index core that is easier to couple but limited by modal bandwidth.
• History: Ibn Sahl described the law geometrically in a treatise on burning instruments in the tenth century, according to Roshdi Rashed's 1990 study; Willebrord Snell found it in 1621 but never published; Descartes published it in La Dioptrique in 1637.
• Fibre taps are physically possible because bending a fibre lets some light escape. Encryption, not the glass, protects the data.

What is refraction, and what is the refractive index?

Refraction is the change in direction of a wave as it crosses from one medium into another in which it travels at a different speed. HyperPhysics puts it plainly: light passing from a fast medium into a slow one bends toward the normal, the imaginary line perpendicular to the boundary. Light passing from slow to fast bends away from the normal.

Why does a change of speed produce a change of direction? Picture a wide wavefront hitting a glass surface at an angle. One edge of the wavefront reaches the glass first and slows down while the other edge is still travelling at full speed in air. The front pivots, the same way a column of marchers turns when the people on one side step onto mud. Huygens' principle formalises this: every point on a wavefront acts as a source of new wavelets, and the new front is the envelope of those wavelets. Wavelets that start inside the glass grow more slowly, so the front tilts.

The speed drops and the wavelength shortens in proportion, but the frequency, set by the source, stays the same. A 1550 nm telecom signal has a wavelength of about 1073 nm inside silica (1550 divided by 1.444) at an unchanged frequency of about 193 THz. Fibre engineers quote vacuum wavelengths because they label the light, not its length in the glass.

The refractive index

The refractive index n is the ratio of the speed of light in vacuum to the speed of light in the material:

n = c / v

The speed of light in vacuum is exact by definition. NIST's CODATA entry gives c = 299 792 458 m/s with "exact" in the uncertainty field, because, as NIST explains, the metre was redefined by international agreement in 1983 as the distance light travels in vacuum in 1/299,792,458 of a second, which locked the value of c. So the index is a pure number, always at least 1 for ordinary transparent materials.

The table uses HyperPhysics values except fused silica, which we computed from Malitson's formula. HyperPhysics notes that its values are approximate and do not account for dispersion, the small change of index with wavelength.

Dispersion: the index depends on colour

Dispersion is why a prism splits white light. For fused silica, Malitson's 1965 measurements, fitted to a three term Sellmeier equation, give an index of 1.4584 at 589.3 nm, 1.4525 at 850 nm, 1.4468 at 1310 nm and 1.4440 at 1550 nm. The index falls as the wavelength rises across the visible and near infrared. In fibre this has a second consequence, chromatic dispersion, which spreads a pulse because its different wavelengths travel at slightly different speeds. ITU-T G.652 specifies it in picoseconds per nanometre per kilometre, and we come back to it below.

Water is similar. The International Association for the Properties of Water and Steam (IAPWS) publishes a formulation, release R9-97, valid from 0.2 to 1.1 µm. Its verification table lists an index of 1.334344 at 589 nm and 0 °C at 0.1 MPa, and 1.326135 at 1014 nm under the same conditions. At room temperature the value is close to the 1.333 that textbooks use.

What does Snell's law say?

Snell's law relates the angles on both sides of a boundary:

n₁ sin θ₁ = n₂ sin θ₂

Here θ₁ is the angle of incidence and θ₂ the angle of refraction, both measured from the normal, not from the surface. The incident ray, the refracted ray and the normal all lie in one plane.

A few consequences fall straight out of the equation:

1. At normal incidence nothing bends. If θ₁ = 0, sin θ₁ = 0, so θ₂ = 0.
2. Into a higher index, the ray bends toward the normal. If n₂ is greater than n₁, sin θ₂ must be smaller than sin θ₁.
3. Into a lower index, the ray bends away from the normal, and there is an angle at which the refracted ray would have to travel along the surface. That angle is the critical angle.
4. The path is reversible. Swap the two sides and the same ray retraces itself.

There is also a speed form. Since n = c/v, the law can be written sin θ₁ / v₁ = sin θ₂ / v₂. This version makes the physical cause obvious: the ratio of the sines equals the ratio of the speeds.

Why the law has this form

Two classic derivations give the same result. The wave picture, already sketched above, matches the wavefront along the boundary: the spacing of crests along the surface must be the same on both sides, which gives λ₁ / sin θ₁ = λ₂ / sin θ₂, and since λ = v/f with the same frequency on both sides, you get Snell's law. The second comes from Pierre de Fermat. MacTutor's biography records that Fermat was unhappy with Descartes' derivation and later deduced the same sine law from a single principle about the path light follows, now known as Fermat's principle: light travels between two points along the path of least time. A lifeguard running on sand and swimming in water reaches a swimmer fastest by running a bit further on sand, where she is faster, and entering the water at an angle. Minimising the total time with calculus gives exactly sin θ₁ / v₁ = sin θ₂ / v₂.

Maxwell's equations give the deepest version: matching the fields at the boundary reproduces Snell's law and also gives the Fresnel equations for how much light reflects.

Who discovered the law of refraction, and when?

The law carries a Dutch name, but its history runs from tenth century Baghdad through seventeenth century Leiden and Paris.

Ibn Sahl

Roshdi Rashed's 1990 paper in Isis, "A Pioneer in Anaclastics: Ibn Sahl on Burning Mirrors and Lenses", argued from a reconstructed manuscript that Ibn Sahl, working in Baghdad in the late tenth century, used a relationship equivalent to the law of refraction to design lenses that focus light without spherical aberration. Anaclastics is the old name for the study of refraction. Ibn Sahl did not write the law in trigonometric form; he expressed it as a fixed ratio between line segments in a geometric figure, which amounts to the same thing. We confirmed the paper's bibliographic details through Crossref; the full text sits behind the publisher's paywall, so the summary here follows the paper's title and the way it is widely cited rather than a fresh reading of every page.

Snell and Descartes

The MacTutor History of Mathematics archive at the University of St Andrews records that Willebrord Snell (1580 to 1626), a Leiden mathematician best known for pioneering triangulation in surveying, discovered the law in 1621. He never published it. His result became known only in 1703, when Huygens printed it in Dioptrica. MacTutor's biography notes that Snell had studied the optics literature closely, including Ibn al-Haytham's experiments, and that he first found the law in a secant form rather than the sine form taught today.

René Descartes published the sine law in 1637 in La Dioptrique, one of three appendices to the Discourse on Method printed at Leiden. MacTutor observes that Descartes did not cite earlier work. Whether he knew of Snell's manuscript has been argued for centuries without a settled answer, which is why French textbooks often say "loi de Snell-Descartes".

Worked example 1: light entering water and glass

Take a ray of yellow light in air striking a flat surface at 30° from the normal. Use n_air = 1.000277, n_water = 1.333 and n_silica = 1.4584.

Into water.

sin θ₂ = (n_air / n_water) × sin 30° = (1.000277 / 1.333) × 0.5 = 0.37520

θ₂ = arcsin(0.37520) = 22.04°

Into fused silica.

sin θ₂ = (1.000277 / 1.4584) × 0.5 = 0.34294

θ₂ = arcsin(0.34294) = 20.06°

The higher index bends the ray more. Running the same calculation for steeper angles shows how the bending grows:

Even at a grazing 80°, light enters the water at less than 48°. A diver looking up therefore sees the whole sky squeezed into a cone about 97° across, known as Snell's window, whose half-angle is the critical angle for water.

The reverse direction. A ray going from water into air at 30° from the normal leaves at:

sin θ₂ = (1.333 / 1.000277) × 0.5 = 0.66632, so θ₂ = 41.78°

It bends away from the normal. This is why a pool looks shallower than it is: rays from the bottom bend away from the normal as they leave, so your eye traces them back to a point that is higher than the real floor. For a viewer looking almost straight down, the apparent depth is close to the real depth divided by 1.333, so a 2.0 m pool looks about 1.5 m deep.

Worked example 2: the critical angle and total internal reflection

When light goes from a higher index n₁ to a lower index n₂, the refracted angle grows faster than the incident angle. At some incident angle the refracted ray would have to skim along the surface, θ₂ = 90°. Setting sin θ₂ = 1 in Snell's law gives the critical angle:

sin θc = n₂ / n₁, so θc = arcsin(n₂ / n₁)

For incidence beyond θc, Snell's law has no real solution, because it would demand a sine larger than 1. No light is transmitted and the boundary reflects all of it. HyperPhysics describes the condition directly: for incident angles greater than the critical angle there will be total internal reflection.

Glass to air. For fused silica at 589 nm:

sin θc = 1.000277 / 1.4584 = 0.68587

θc = arcsin(0.68587) = 43.30°

At 1550 nm, where the silica index is 1.4440, θc is 43.84°. So any ray inside a block of fused silica hitting its surface at more than about 43° to 44° from the normal stays inside.

Water to air. sin θc = 1.000277 / 1.333 = 0.75039, so θc = 48.62°. Double it and you get the 97° Snell's window from the previous section.

Glass to water. sin θc = 1.333 / 1.4584 = 0.91401, so θc = 66.07°. A glass to water boundary traps light only at much steeper angles, because the index step is smaller. This is the first hint at how a fibre works: the cladding around a fibre's core has an index only slightly lower than the core, so the critical angle inside the fibre is very large, above 80°.

What happens near and beyond the critical angle

Below θc, part of the light reflects and part transmits. The reflected fraction at normal incidence is R = ((n₁ − n₂)/(n₁ + n₂))². For silica and air that is (0.4584/2.4584)² = 3.5 percent, which is why a fibre connector end face that is cut flat and left with an air gap reflects a few percent of the signal back toward the laser. For diamond it is 17 percent, one reason diamonds look bright. The Fresnel equations give the angle dependent version; the reflected fraction climbs steeply as θ₁ approaches θc and reaches 100 percent at θc.

Beyond θc the reflection is total, but the field does not stop dead at the boundary. A wave called the evanescent wave extends a short distance into the lower index medium and decays exponentially, carrying no net energy away as long as nothing is there to absorb it. Its penetration depth, the distance over which its intensity falls by a factor of e, is roughly:

d = λ / (2π √(n₁² sin² θ₁ − n₂²))

For 1550 nm light in silica (n = 1.444) hitting an air boundary at 60°, d = 1550 / (2π × √(1.444² × 0.75 − 1)) = 1550 / (2π × 0.751) ≈ 329 nm. At 80° it is about 244 nm, and at 50°, closer to the critical angle, about 522 nm. If another high index material comes within a few hundred nanometres of the surface, the evanescent wave can couple into it and carry energy across the gap. That effect, called frustrated total internal reflection, comes back in the sections on fingerprint sensors and fibre taps.

Table: refractive index and critical angle against air

The table below gives the critical angle against air (n = 1.000277) and, where the material's index is above water's, against water. Values are for yellow light near 589 nm unless noted, and are approximate because index varies with wavelength and temperature.

The higher the index, the smaller the critical angle. Immersion in water raises every critical angle, which is why a wet or greasy diamond loses much of its sparkle.

How does an optical fibre guide light?

An optical fibre is a thin strand of glass made of two concentric layers. The inner core has a slightly higher refractive index than the surrounding cladding. Light launched into the core at a shallow enough angle hits the core to cladding boundary beyond the critical angle and is totally internally reflected, again and again, all the way down the fibre. HyperPhysics summarises it: once the light is introduced within the numerical aperture of the fiber, it continues to reflect almost losslessly off the walls and can travel long distances.

For standard telecom fibre, both layers are silica glass. The core is doped, typically with germanium dioxide, to raise its index slightly. The cladding is 125 µm in diameter; ITU-T G.652 gives "the recommended nominal value of the cladding diameter is 125 µm", with a tolerance of 0.7 µm for the G.652.D subcategory. Outside the cladding, a polymer coating protects the glass mechanically. The coating does not guide light; the core and cladding do.

Why not use air as the cladding?

A bare glass rod in air does guide light, but its guiding surface is then the outside of the rod, so every fingerprint, scratch or support frustrates the reflection and leaks light. A glass cladding moves the guiding boundary inside the fibre, where nothing can touch it, and the evanescent field dies away within the cladding.

A small index step also has a second benefit: it limits how many paths, or modes, the core can carry, which is the key to single mode operation.

Kao's insight: purity, not geometry

Total internal reflection was well understood in the nineteenth century. What held fibre back was absorption and scattering in the glass itself. According to the Nobel Prize committee's 2009 press release, fibres available in the 1960s could carry light only about 20 metres before most of it was lost. In 1966 Charles Kao calculated that a fibre of very pure glass could carry signals 100 kilometres. The first ultrapure fibre was made four years later, in 1970, and Kao shared the 2009 Nobel Prize in Physics for the work. Today's G.652.D cable is specified at no more than 0.30 dB/km between 1530 and 1565 nm, which means half the light survives about 10 km.

What are numerical aperture and acceptance angle?

A fibre only guides rays that meet the core to cladding boundary beyond the critical angle. Trace that requirement back to the fibre's end face and you get a cone of acceptance: rays inside the cone are trapped; rays outside it leak into the cladding and are lost.

The numerical aperture measures the cone. ITU-T G.651.1 defines it as "the sine of the vertex half-angle of the largest cone of rays that can enter or leave the core", multiplied by the index of the medium outside the fibre. For a step index fibre, applying Snell's law at the end face and the critical angle condition at the core to cladding boundary gives:

NA = n₀ sin θa = √(n₁² − n₂²)

where n₀ is the index outside (1 for air, near enough), θa is the acceptance half-angle, n₁ is the core index and n₂ the cladding index.

Derivation in four lines

1. At the end face, a ray enters from air at θa and refracts to angle θt from the fibre axis: sin θa = n₁ sin θt.
2. Inside, the ray meets the core wall at angle 90° − θt from the wall's normal.
3. For total internal reflection, sin(90° − θt) = cos θt must be at least n₂/n₁.
4. At the limit, sin θt = √(1 − (n₂/n₁)²), so sin θa = n₁ √(1 − (n₂/n₁)²) = √(n₁² − n₂²).

A related quantity, the relative index difference, is Δ = (n₁ − n₂)/n₁. For small Δ, NA ≈ n₁ √(2Δ).

Worked example 3: the G.651.1 multimode fibre

ITU-T G.651.1 (11/2018), the recommendation for 50/125 µm graded index multimode fibre in access networks, gives a nominal numerical aperture of 0.20 with a tolerance of 0.015, measured at 850 nm.

Acceptance angle. sin θa = 0.20, so θa = arcsin(0.20) = 11.54°. The full acceptance cone is about 23.1° across. Light from a source that spreads wider than that does not stay in the core.

Implied core index. Take the cladding as pure fused silica at 850 nm, n₂ = 1.4525 from Malitson's formula. Then n₁ = √(n₂² + NA²) = √(1.4525² + 0.20²) = 1.4662. The relative index difference is Δ = (1.4662 − 1.4525)/1.4662 = 0.93 percent. A difference of under one percent in refractive index is all it takes to guide light.

Critical angle inside the core. sin θc = n₂/n₁ = 1.4525/1.4662, so θc = 82.16°. Guided rays meet the wall at more than 82° from its normal, which means they travel within 7.8° of the fibre axis. The rays inside a fibre do not zigzag steeply; they graze the walls.

One subtlety the recommendation spells out: the measured NA "is about 5% lower" than the maximum theoretical NA computed from a refractive index profile, because the measurement uses far field light rather than an ideal geometric cone. With a measured 0.20, the theoretical value is about 0.21. For a graded index core, n₁ is the peak index at the centre of the core.

Worked example 4: a single mode fibre (illustrative values)

G.652 does not specify numerical aperture; it specifies mode field diameter instead, because in a single mode fibre the ray picture stops being accurate. For an illustrative calculation, assume a step index single mode fibre with NA = 0.12 and a core radius a = 4.1 µm, values typical of the category but not taken from the standard.

Acceptance half-angle: arcsin(0.12) = 6.89°, much narrower than the multimode fibre's 11.5°. With cladding n₂ = 1.4440 at 1550 nm, the core index is √(1.4440² + 0.12²) = 1.4490 and Δ = 0.34 percent.

Whether such a fibre carries one mode or many is decided by the normalised frequency, or V number:

V = (2π a / λ) × NA

Waveguide theory for a step index fibre shows that only the fundamental mode propagates when V is below about 2.405, the first zero of a Bessel function that appears in the mode equations; the RP Photonics encyclopedia, a secondary reference, gives the same threshold. For our illustrative fibre:

The wavelength at which V equals 2.405 is the cutoff wavelength: λc = 2π a NA / 2.405 = 1.285 µm. ITU-T G.652.D requires the cable cutoff wavelength to be no more than 1260 nm. Cabled fibre cuts off at a shorter wavelength than a short straight piece of the same fibre, because bends in the cable strip the weakly guided second mode, which is why a fibre whose theoretical cutoff is slightly above 1260 nm can still meet a 1260 nm cable cutoff. The point of the exercise is the shape of the trade off, not the exact numbers: a smaller core or a lower NA moves the cutoff to shorter wavelengths, and a larger core moves it longer.

Why do single mode and multimode fibres differ?

The difference comes down to how many distinct paths, or modes, the core supports, and what that does to a pulse.

Modal dispersion

In a multimode step index fibre, a ray going straight down the axis travels a distance L. A ray at the critical angle bounces its way down a longer path, L × n₁/n₂. Both travel at the same speed in the core, c/n₁, so the steep ray arrives later. The spread in arrival time is:

Δt = (L n₁ / c) × (n₁ − n₂)/n₂

Worked example 5. For the step index equivalent of our G.651.1 fibre over 1 km: Δt = (1000 × 1.4662 / 299 792 458) × (0.0137 / 1.4525) ≈ 46 ns. A 1 ns pulse comes out smeared to nearly 50 ns. If you send bits faster than one every 46 ns or so, about 20 Mbit/s for a kilometre, adjacent pulses overlap. That is modal dispersion, and it is why step index multimode fibre is no longer used for data links.

Graded index cores. The fix for multimode fibre is to grade the core's index, highest at the centre and falling smoothly toward the cladding. Rays that wander away from the axis spend more of their time in lower index glass, where light travels faster, which compensates for their longer path. Simple ray analyses of an ideal profile suggest the spread can fall by a factor of roughly Δ/8 compared with the step index case, from tens of nanoseconds to tens of picoseconds per kilometre, though real fibres fall short of the ideal. G.651.1 captures the result as a minimum modal bandwidth length product of 500 MHz·km at both 850 and 1300 nm for an overfilled launch, enough, per its summary, for 1 Gbit/s Ethernet over links up to 550 m using 850 nm transceivers.

Single mode. Shrink the core until V is below 2.405 and only one mode exists. Modal dispersion disappears entirely. What remains is chromatic dispersion, the spread caused by the different wavelengths in a laser's output travelling at slightly different speeds, plus polarisation mode dispersion. G.652 fibre has its zero dispersion wavelength near 1310 nm; the recommendation bounds it between 1300 and 1324 nm for G.652.D, and limits dispersion at 1550 nm to between 13.3 and 18.6 ps/(nm·km). Modern coherent transceivers compensate chromatic dispersion digitally, as covered in our optical transceiver post.

Comparison table: single mode versus multimode

What the ITU-T recommendations specify

G.652, characteristics of a single mode optical fibre and cable. First created in 1984, according to the summary of the current 08/2024 edition, which describes a fibre with "zero-dispersion wavelength around 1310 nm" that was "originally optimized for use in the 1310 nm wavelength region but can also be used in the 1550 nm region". The current edition lists two subcategories in its tables, G.652.B and G.652.D, the latter being the reduced water peak fibre whose attenuation is controlled across the whole band from 1310 to 1625 nm, including at 1383 nm after a hydrogen ageing test. Beyond the values in the table above, its Table 2 sets a mode field diameter tolerance of 0.4 µm, core concentricity error of at most 0.6 µm, macrobending loss of at most 0.1 dB for 100 turns at 30 mm radius measured at 1625 nm, proof stress of at least 0.69 GPa and a maximum PMDQ of 0.20 ps/√km.

G.657, characteristics of a bending loss insensitive single mode optical fibre and cable. The 08/2024 edition is the fifth. Its summary explains that access networks, crowded telecom offices and data centres bend fibre tightly and handle it often. Category A fibres are fully compliant with G.652.D and can be used anywhere G.652 can; category B fibres trade some G.652 compatibility for very low loss at very small radii. The recommendation gives minimum design radii of 10 mm for G.657.A1, 7.5 mm for G.657.A2 and 5 mm for G.657.B3. This edition merged the former category B2 into A2.

G.651.1, 50/125 µm multimode graded index fibre for the optical access network. The current 11/2018 edition is the second. It describes an improved version of the 50/125 µm fibre in the older G.651, aimed at multi tenant buildings delivering broadband to individual apartments; its Table 1 supplies the multimode values above.

The IEC 60793 family. The ITU-T recommendations refer to IEC 60793 for test methods and product specifications. G.652, for example, points to IEC 60793-2-50 for the hydrogen ageing type test on its B-652.D fibre category, and G.651.1 points to IEC 60793-1-20 for measuring core diameter. IEC documents are sold rather than published free, so we rely on what the ITU-T texts say about them.

Why bending matters: the link to total internal reflection

A bend changes the geometry. On the outside of a curve, rays meet the core to cladding boundary at a smaller angle from the normal, and some fall below the critical angle and escape. In wave terms, the part of the mode's field on the outside of the bend would have to travel faster than light in the cladding to keep up, so it radiates away. Bend loss rises steeply as the radius shrinks and as the wavelength grows, which is why G.652.D tests macrobending at 1625 nm, the longest wavelength in its band. Bend insensitive G.657 fibres reduce the loss by raising the index contrast near the core or adding a lower index trench in the cladding, so the light is held more tightly.

This is also the physics behind a security consideration: light that leaves a bent fibre can be collected.

Where else does total internal reflection show up?

Prisms

A right angled glass prism turns a beam through 90° or 180° with total internal reflection instead of a mirror. Light enters one face at normal incidence, meets the long face at 45°, which is beyond the roughly 41° critical angle of common glass, and reflects totally. HyperPhysics notes that total internal reflection is also used in polarising prisms. A prism reflector has no metal coating to tarnish, which is why binoculars use prisms.

Lenses

Lenses rely on refraction rather than total internal reflection, but Snell's law sets every surface. Ibn Sahl's tenth century problem, which lens shape brings parallel rays to a single point, is the same one lens design software solves today by applying Snell's law at every surface for millions of rays.

Mirages

Air's refractive index depends on its density, so a layer of hot air over a road has a slightly lower index than the cooler air above it. A ray from the sky heading down toward the road at a very shallow angle meets this gradient and bends back up, effectively reflecting. Your eye sees the sky on the road and reads it as water.

Worked example 6. Take air at 20 °C with n = 1.000277 and a layer next to the road at 50 °C. Since n − 1 is roughly proportional to air density, and density at constant pressure is inversely proportional to absolute temperature, the hot layer's index is about 1 + 0.000277 × 293.15/323.15 = 1.000251. The critical angle condition gives the grazing angle at which reflection becomes total: cos α = 1.000251/1.000277, so α ≈ 0.41° above the horizontal. Only rays travelling almost parallel to the road are reflected, which is why a road mirage appears only in the distance and vanishes as you approach. For precise work, NIST's Engineering Metrology Toolbox provides a calculator for the index of air based on the Ciddor and modified Edlén equations, which account for temperature, pressure, humidity and carbon dioxide.

Diamonds

Diamond's index of 2.417 gives it a critical angle against air of just 24.4°. A brilliant cut diamond is shaped so that light entering the top bounces off the angled lower facets beyond the critical angle and returns out through the top. Strong dispersion splits the returning light into flashes of colour called fire.

Fingerprint sensors

Many optical fingerprint readers use frustrated total internal reflection. Light inside a glass or plastic prism strikes the top surface beyond the critical angle and would reflect totally to a camera. Where the ridges of a finger touch the surface, the skin sits well within the few hundred nanometre reach of the evanescent wave and absorbs or scatters the light, so ridges show as dark lines on a bright background. The valleys, separated from the glass by air, leave the reflection total.

Why are fibre taps physically possible?

Total internal reflection traps light only while its conditions hold: the right angle and a lower index region just outside the core. Change either and some light escapes. Conceptually, there are two routes:

• Bending. Bend a fibre tightly enough and part of the guided light falls below the critical angle at the outer side of the bend, or radiates in the wave picture. That light leaves the cladding and can be collected by a photodetector held against the bend. This is the same macrobend loss that G.652 and G.657 limit.
• Evanescent coupling. Bring another high index material within the reach of the evanescent field, typically after removing protective layers, and some light couples across. This is the same frustrated total internal reflection used deliberately in fibre couplers.

In both cases only a small fraction of the light needs to be diverted for a sensitive receiver to read the signal, and the main signal carries on. The tapping does leave a physical trace, a small extra loss at one point, which is what optical power monitoring and optical time domain reflectometers measure. We describe the mechanism only at this level; this post does not give step by step instructions.

The lessons for defenders are practical:

1. Encrypt the data, not just the path. Fibre's physics makes tapping harder than on copper, but not impossible. Link layer encryption on Ethernet (MACsec, the IEEE 802.1AE standard for Media Access Control Security, whose current base revision is 802.1AE-2018), IPsec or TLS on top makes a tapped signal useless to the person collecting it.
2. Baseline your links. Record the loss and reflection trace of each important span when it is installed. A new loss event at a fixed distance is worth investigating even if the link still works.
3. Watch received optical power. Transceiver diagnostics report it; a sudden step change deserves a look, though routine maintenance causes the same symptom.
4. Protect the physical route. Locked enclosures, controlled patch panels and conduit for sensitive runs reduce opportunity.

Walkthrough: reproduce every number in this post

You can check each calculation, and experiment with your own materials, with a short Python script. It uses only the standard library, so it has no third party dependencies to drift.

Step 1: create a clean environment

python3 -m venv .venv
source .venv/bin/activate
python -m pip install --upgrade pip
python --version

If you later add libraries, pin them so the script gives the same answer next year:

python -m pip install numpy matplotlib
python -m pip freeze > requirements.txt
python -m pip install -r requirements.txt

Step 2: the optics calculations

Save this as snell.py:

import math

C = 299_792_458          # m/s, exact (NIST CODATA)
deg, rad = math.degrees, math.radians

def silica(um):
    """Fused silica index, Malitson (1965) Sellmeier fit, 0.21 to 6.7 um."""
    l2 = um * um
    return math.sqrt(1
        + 0.6961663 * l2 / (l2 - 0.0684043**2)
        + 0.4079426 * l2 / (l2 - 0.1162414**2)
        + 0.8974794 * l2 / (l2 - 9.896161**2))

def refract(n1, n2, theta1_deg):
    s = n1 * math.sin(rad(theta1_deg)) / n2
    return None if s > 1 else deg(math.asin(s))   # None means total internal reflection

def critical(n1, n2):
    return deg(math.asin(n2 / n1)) if n1 > n2 else None

AIR, WATER = 1.000277, 1.333
for wl in (0.5893, 0.85, 1.31, 1.55):
    print(f"silica n at {wl} um: {silica(wl):.4f}")

n_si = silica(0.5893)
for t in (30, 45, 60, 80):
    print(f"{t:2d} deg in air -> water {refract(AIR, WATER, t):.2f}, silica {refract(AIR, n_si, t):.2f}")

print("water to air, 30 deg:", round(refract(WATER, AIR, 30), 2))
print("critical silica/air:", round(critical(n_si, AIR), 2))
print("critical water/air: ", round(critical(WATER, AIR), 2))
print("critical diamond/air:", round(critical(2.417, AIR), 2))

# Numerical aperture: G.651.1 multimode fibre, NA 0.20 at 850 nm
NA = 0.20
n2 = silica(0.85)
n1 = math.sqrt(n2**2 + NA**2)
print(f"acceptance half-angle {deg(math.asin(NA)):.2f} deg, core n1 {n1:.4f}, "
      f"Delta {(n1 - n2) / n1 * 100:.2f} %, core critical {critical(n1, n2):.2f} deg")

# Step index modal spread over 1 km
dt = 1000 * n1 / C * (n1 - n2) / n2
print(f"step index modal spread: {dt * 1e9:.1f} ns/km")

# Single mode, illustrative a = 4.1 um, NA = 0.12
a, na_sm = 4.1, 0.12
for wl in (1.26, 1.31, 1.55):
    print(f"V at {wl} um: {2 * math.pi * a * na_sm / wl:.2f}")
print(f"cutoff: {2 * math.pi * a * na_sm / 2.405:.3f} um")

# Evanescent penetration depth, silica/air at 1550 nm
n = silica(1.55)
for t in (50, 60, 80):
    d = 1550 / (2 * math.pi * math.sqrt((n * math.sin(rad(t)))**2 - 1))
    print(f"penetration depth at {t} deg: {d:.0f} nm")

Run it:

python snell.py

You should see a silica index of 1.4584 at 0.5893 µm and 1.4440 at 1.55 µm, refraction angles of 22.04° and 20.06° at 30°, critical angles of 43.30°, 48.62° and 24.45°, an acceptance half-angle of 11.54° with a core index of 1.4662, a modal spread near 46 ns per kilometre, V numbers of 2.45, 2.36 and 1.99, and penetration depths of about 522, 329 and 244 nm.

Step 3: measure Snell's law yourself

You can check the law with a laser pointer, a clear rectangular container of water, a protractor printed on paper and a little milk or powdered coffee creamer in the water to make the beam visible.

1. Tape the protractor to the side of the container with its centre on the water line. The 0° line is the normal.
2. Shine the beam down into the water through the centre of the protractor at 30°, 45° and 60° from the normal.
3. Read the angle of the beam inside the water each time.
4. Compute sin θ₁ / sin θ₂ for each pair. It should come out close to 1.33 for every angle, which is the point: the ratio of sines, not the ratio of angles, stays constant.
5. Now aim the beam up from below through the side wall at steeper and steeper angles to the water surface. Somewhere near 49° from the normal the beam stops emerging and the surface becomes a mirror.

Never look into the beam or point it at anyone, and use a low power pointer.

Step 4: keep the tooling healthy

The script above has no third party dependencies, which is deliberate. Real optical design tools, network monitoring dashboards and transceiver management services are another story: they pull large open source trees. Before you share a script that has grown dependencies, or ship a service built from one, scan the folder for known vulnerable components:

safeguard scan --dir .

Results are written as JSON under ./output.

Common mistakes

Measuring angles from the surface instead of the normal. Snell's law uses angles from the normal. A ray "at 30° to the water surface" is at 60° from the normal, and putting 30° into the formula gives the wrong answer.

Forgetting that total internal reflection only works from high index to low. There is no critical angle going from air into glass. If n₂/n₁ is greater than 1, arcsin has no solution and your calculator will return an error, which is the correct answer.

Assuming the index is a single number. Index depends on wavelength, temperature and, for gases, pressure. Silica is 1.458 in yellow light and 1.444 at 1550 nm. Using the visible value for a telecom calculation shifts the critical angle by half a degree and the core index you infer from an NA by more than the core to cladding difference you are trying to compute.

Treating numerical aperture as an angle. NA is a sine, not an angle. An NA of 0.20 is an acceptance half-angle of 11.5°, not 20° or 0.20 radians.

Confusing core diameter with mode field diameter. Single mode fibre specifications quote mode field diameter, the width of the light's intensity profile, which is larger than the physical core because part of the light travels in the cladding. G.652.D's 8.6 to 9.2 µm figure is a mode field diameter.

Mixing single mode and multimode fibre. Connecting a 9 µm single mode transceiver to 50 µm multimode fibre, or the reverse, causes large losses or modal noise. The cladding diameter is the same 125 µm, so the connectors fit, which is what makes the mistake easy.

Ignoring minimum bend radius. G.652.D fibre is tested at 30 mm radius. Pulling it around a 10 mm corner in a crowded rack can add loss, especially at 1550 and 1625 nm. If routing is tight, specify G.657 fibre.

Believing fibre is untappable. Fibre is harder to tap than copper and taps leave traces, but the physics allows it. Treat the link as untrusted and encrypt.

FAQ

What is Snell's law in simple terms?

When light crosses from one transparent material into another, it bends by an amount set by the two materials' refractive indices. The rule is n₁ sin θ₁ = n₂ sin θ₂, with angles measured from the line perpendicular to the surface. Going into a denser material, light bends toward that line; going out, it bends away.

What is the refractive index of glass and water?

Water is about 1.333 at room temperature in yellow light. Common crown glass ranges from about 1.50 to 1.62 depending on composition. Fused silica, the glass in optical fibre, is 1.4584 at 589 nm and 1.4440 at 1550 nm, from Malitson's formula.

How do you calculate the critical angle?

Use θc = arcsin(n₂/n₁), where n₁ is the index of the material the light is in and n₂ the index of the material outside. For glass with n = 1.5 into air, θc = arcsin(1/1.5) = 41.8°. It only exists when n₁ is larger than n₂.

What is total internal reflection?

It is the complete reflection of light at a boundary with a lower index material when the angle of incidence exceeds the critical angle. No light is transmitted, though an evanescent field extends a few hundred nanometres past the boundary.

How does total internal reflection work in optical fibre?

The core's index is slightly higher than the cladding's, by under one percent. Light entering within the acceptance cone meets the boundary beyond the critical angle, above 80°, and reflects totally at every bounce.

What is numerical aperture in fibre optics?

It is the sine of the half-angle of the cone of light a fibre accepts, NA = √(n₁² − n₂²). The ITU-T G.651.1 multimode fibre has a nominal NA of 0.20, which is an acceptance half-angle of about 11.5°.

What is the difference between single mode and multimode fibre?

Single mode fibre has a core small enough, around 9 µm of mode field diameter, that only one light mode propagates, so there is no modal dispersion and it can carry signals for tens of kilometres. Multimode fibre, such as the ITU-T G.651.1 type, has a 50 µm core that is easier to couple light into but suffers modal dispersion, limiting it to links of hundreds of metres at high speeds.

Who discovered Snell's law?

Ibn Sahl described an equivalent geometric law in Baghdad around 984, according to Roshdi Rashed's 1990 study. Willebrord Snell found it in 1621 but did not publish; Huygens made his work public in 1703. René Descartes published the sine law in 1637 in La Dioptrique.

Can optical fibre be tapped?

Yes, in principle. Bending a fibre or bringing another material close to the core lets a small fraction of the light escape, which a sensitive receiver can read. Taps add a small, measurable loss, and encrypting the traffic with MACsec, IPsec or TLS makes tapped light useless.

The software around the glass: an honest bridge

Snell's law is physics, and Safeguard does not do physics. Its models are single purpose security models for software; they do not calculate refraction, design fibre or analyse optical hardware, and nothing in this post was computed by them. The connection is narrower and real. The networks that run on G.652 fibre are managed by software: network operating systems on switches and routers, management software for transceivers and optical line systems, link monitoring dashboards, and the Python and web services that collect received power readings and loss traces. That software is built from open source dependencies, and much of it is ordinary code your team writes, packages into container images and ships. Where Safeguard helps is there: scanning those repositories, images and SBOM files for known vulnerable open source components, showing which projects carry a given version, and, when a fixed version exists, opening an upgrade pull request for review. It does not scan transceiver firmware or produce hardware bills of materials.

The glass does not need patching, but the code that watches it does: monitoring services, network automation scripts and management dashboards pull open source dependencies again every time they are rebuilt. The fit is indirect for the optics, and for the software your team builds around a fibre network, the same dependency hygiene applies, as the next section shows.

Fix it without waiting for upstream

Whatever the vulnerability, most of the risk sits in two gaps. The first is the time between a fix being published and that fix actually running in your production builds. The second, and worse, is the time before any fix exists at all, while a maintainer works out a patch and a release schedule. Waiting through both gaps is a habit, not a requirement. Safeguard gives teams three ways to close them.

1. Autonomous remediation: pull requests, not tickets

When a fixed version exists, Safeguard's Griffin models open a tested, policy gated pull request that moves the affected dependency to the safe version. You can run it on one repository or in bulk across every project that carries the vulnerable version. Remediation covers eight categories, including dependencies, licenses, security posture and findings. Reachability analysis ranks the vulnerable code paths your application actually calls first, so the first pull requests you merge are the ones that matter most. Autonomous remediation ships switched off, with every category set to Manual, so your team decides how much runs on its own and what still needs a human approval.

2. Gold Open Source: the safe version in seconds, free

gold.safeguard.sh is free and needs no login. Search any CVE or package to see affected and fixed versions, CVSS, EPSS, CISA KEV status and a recommended safe version, or use the "Am I affected?" checker. The same data is available through a free read API, README status badges and RSS threat feeds. The safeguard-sh/gold-scan-action GitHub Action can fail a build on a known exploited vulnerability, on any known vulnerability, or never, so a version you have just removed cannot quietly come back in next week's merge.

3. Gold Registry: start from components that are already fixed

registry.safeguard.sh provides hardened, drop in replacements for upstream packages and container images. Every Gold artifact is held continuously to zero exploitable critical or high CVEs, zero malware and a completed license check, with SLSA L3 or higher provenance, a Sigstore signature and a public evidence page you can hand to an auditor. An artifact loses its Gold badge if it regresses. Teams can point their own package directory at the registry so hardened equivalents resolve without changing how developers install anything.

When upstream has no fix yet, or will not ship one, Safeguard first works to get the fix accepted upstream. If the issue is severe or upstream will not move, Safeguard builds a fork, fixes the vulnerability there and publishes a hardened Gold build, so you are not stuck on a maintainer's release schedule. You can also request a hardened, compatibility tested build of a specific package. This fork and patch flow is in early access today, so ask which packages it covers before you plan around it.

One honest limit: when the vulnerable code lives in a vendor appliance, a hosted service or an operating system runtime you do not build yourself, the fix still has to come from that vendor or distribution. The steps above apply to the open source dependencies and container images you build and ship, which is where most teams carry the largest and least tracked share of their exposure.

See how Safeguard would help

None of these demos involve optics; they show the software side described in the bridge above.

The first demo is a real Safeguard CLI session: it installs the CLI, opens the interactive terminal and asks it to fix a lodash CVE in a repository. It reads the finding and advisory, asks before each command, edits package.json to lodash 4.17.21, runs npm install and npm test with approval, prints a summary and pushes nothing. The same flow applies to a fibre link monitoring dashboard or a network automation service your team maintains: scan the repository, then let the assistant propose the dependency upgrade under your approval.

Click through it at your own pace, or open it full screen.

The second demo sets up the Safeguard runner on your own machine, approves it with its device code and then scans from the terminal. That suits network engineering teams who keep automation scripts and optical monitoring tools on their own hardware, inside the management network, and want the scan to run where the code lives.

Click through it at your own pace, or open it full screen.

The third demo connects Claude to the Safeguard MCP server with OAuth and lets you choose which tools Claude can use. For an engineer who already works with Claude while writing network tooling, it brings dependency findings into the same conversation, limited to the tools you allow.

Click through it at your own pace, or open it full screen.

To try it on your own repositories, download Safeguard and follow the CLI documentation.

Quick check

Sources

• ITU-T Recommendation G.652 (08/2024): Characteristics of a single-mode optical fibre and cable
• ITU-T Recommendation G.657 (08/2024): Characteristics of a bending-loss insensitive single-mode optical fibre and cable
• ITU-T Recommendation G.651.1 (11/2018): Characteristics of a 50/125 µm multimode graded index optical fibre cable for the optical access network
• NIST CODATA, speed of light in vacuum
• NIST Engineering Metrology Toolbox, Refractive Index of Air (Ciddor and Edlén equations)
• IAPWS R9-97, Release on the Refractive Index of Ordinary Water Substance
• refractiveindex.info, fused silica, Malitson 1965 dispersion formula
• HyperPhysics (Georgia State University), Refraction of Light and Index of Refraction
• HyperPhysics (Georgia State University), Total Internal Reflection
• HyperPhysics (Georgia State University), Fiber Optics
• MacTutor History of Mathematics (University of St Andrews), Willebrord Snell
• MacTutor History of Mathematics (University of St Andrews), René Descartes
• MacTutor History of Mathematics (University of St Andrews), Ibn al-Haytham
• Rashed, R. (1990), A Pioneer in Anaclastics: Ibn Sahl on Burning Mirrors and Lenses, Isis 81(3), 464 to 491
• The Nobel Prize in Physics 2009, press release
• NIST, SI redefinition: meter
• MacTutor History of Mathematics (University of St Andrews), Pierre de Fermat
• MacTutor History of Mathematics (University of St Andrews), Ptolemy
• IEEE 802.1 Working Group, 802.1AE: MAC Security (MACsec)
• RP Photonics Encyclopedia, V number (secondary reference)

Last reviewed: 24 October 2026.