Refraction of Light — Snell's Law

Drop a plastic straw into a glass of nimbu pani on a hot afternoon. Look at it from the side. The straw appears broken at the water line — the part inside the water is shifted sideways from the part in the air, as if two different straws were stuck together at the surface. Now look at a rupee coin lying on the floor of a swimming pool at the Andheri Sports Complex: the coin seems to sit a few centimetres higher than where it actually is, and if you reach in to pick it up, your hand lands on nothing.

Every one of these is light changing direction as it crosses a boundary between two materials with different optical properties. The law governing that direction change is three centuries old, and one line long: n_1\sin\theta_1 = n_2\sin\theta_2. It is called Snell's law after the Dutch astronomer who wrote it down in 1621, though it was known and used by the Persian scholar Ibn Sahl around 984 AD — and the Kerala school of mathematician-astronomers in the 14th century understood the bending of light through water well enough to correct star positions observed through the atmosphere. In this chapter you derive the law from a wave picture, apply it to slabs and pools, and meet the refractive index — the single number that tells you everything about how a medium interacts with light.

The phenomenon — light bends at a boundary

Place a coin at the bottom of an empty tumbler and tip it away from you until the coin just disappears below the rim. Hold that angle. Now, without moving your head, pour water slowly into the tumbler. As the water level rises, the coin reappears — you can see it again, through the water, even though neither your eye nor the coin has moved.

The coin did not move. Your eye did not move. What happened is that the light reaching your eye now travels a different path: it leaves the coin, travels up through the water, bends at the water-air interface, and then continues to your eye. The bend is enough to bring the coin back into the line of sight that was previously blocked by the rim.

Three qualitative facts come out of experiments like this one:

1. The bending happens only at the boundary. Inside either medium the ray travels in a straight line.
2. The angle of bending depends on the angle of incidence. A ray hitting the surface perpendicularly (normal incidence) does not bend at all.
3. The bending is sensitive to what the two media are. Going from air into glass bends the ray more than going from air into water, which bends it more than going from air into alcohol.

The single number that captures "how much medium X bends light" is the refractive index.

Refractive index — how fast light moves

Light travels at c = 2.998 \times 10^8 m/s in vacuum. In air, water, glass, diamond — or any other transparent medium — it travels slower. The refractive index of a medium is the ratio of the vacuum speed to the medium speed:

A table of common values:

Medium	n	Speed v = c/n
Vacuum	1.0000	2.998\times 10^8 m/s
Air (1 atm, 20°C)	1.0003	2.997\times 10^8 m/s
Water (20°C)	1.333	2.250\times 10^8 m/s
Crown glass	1.52	1.972\times 10^8 m/s
Dense flint glass	1.65	1.817\times 10^8 m/s
Diamond	2.42	1.239\times 10^8 m/s

The tiny excess of air's n above 1 is large enough to matter in astronomy — every star you see through the Mount Abu InfraRed Observatory appears a little higher in the sky than it really is, because the atmosphere bends its light downward by a few arcseconds. Near the horizon, the bend is enough that you see the Sun for a minute or two after it has, geometrically, set.

Light slows in a medium — what "slowing" means

A quick clarification: photons still travel at c between atoms even in glass. What the refractive index measures is the group velocity of the electromagnetic wave, which is slower because the photons continually interact with atoms, get absorbed and re-emitted, and the net forward propagation is delayed. You can, for calculation purposes, treat light in a medium as simply moving at c/n — Maxwell's equations inside the medium give exactly this speed, and every optical formula that uses n is consistent with this picture.

Deriving Snell's law from Huygens' principle

The wave picture of reflection (developed in Reflection of Light) extends directly to refraction. The only change is that the wave speed is different on the two sides of the boundary. That one difference produces Snell's law — all of it, in five lines of geometry.

Step 1. A plane wavefront AB in medium 1 hits the boundary obliquely. At time t = 0, point A just touches the boundary; point B is still above it.

Step 2. Let \tau be the time for B to cover the remaining distance BB' to the boundary. Travelling at v_1 = c/n_1:

During this time, A has started radiating a secondary wavelet into medium 2, where the speed is v_2 = c/n_2. The wavelet has grown to radius

Why: Huygens says every point on a wavefront becomes a source of secondary wavelets. Those wavelets spread at the local speed of light in whichever medium they are in. A has been emitting into medium 2 for time \tau, so its wavelet has radius v_2\tau in medium 2.

Step 3. The new wavefront in medium 2 is the tangent from B' to the circle centred at A with radius r. Call the tangent point T. Then AT = r and B'T is perpendicular to AT (the definition of tangent).

Step 4. Work out the geometry. In the right triangle AB'B above the boundary, AB' lies on the boundary, and \angle ABB' = \theta_1 (where \theta_1 is the angle between the incident ray and the normal — and the incident ray is perpendicular to the wavefront AB, so the wavefront itself makes angle \theta_1 with the boundary).

Substituting BB' = v_1\tau:

Step 5. In the right triangle AB'T below the boundary, AT = r is perpendicular to B'T, and \angle TAB' = \theta_2 (the angle of refraction — the refracted ray, perpendicular to the new wavefront B'T, makes angle \theta_2 with the normal).

Substituting AT = v_2\tau:

Step 6. Divide the two equations to eliminate AB'\tau:

Why: the \tau cancels, the AB' cancels, and we are left with a relation between angles and speeds alone.

Step 7. Convert speeds to refractive indices using v = c/n:

Rearranging:

Why: this is Snell's law in its standard form. The quantity n\sin\theta is conserved across the boundary — it does not matter which medium you are in, this combination stays constant.

Step 8. Which way does the ray bend? Going into a denser medium (n_2 > n_1), the ratio \sin\theta_1/\sin\theta_2 = n_2/n_1 > 1, so \sin\theta_2 < \sin\theta_1, so \theta_2 < \theta_1. The ray bends toward the normal. Going into a rarer medium is the reverse — the ray bends away from the normal.

This is the single most used fact in optical design: every pair of glass surfaces in a camera lens, every gemstone cut, every fibre optic cable, every contact lens prescription — all reduce to repeated applications of n_1\sin\theta_1 = n_2\sin\theta_2.

Interactive — drag the angle, watch the bending

Refraction through a parallel-sided slab

Now apply Snell's law twice: once when a ray enters a glass slab, again when it leaves. A plane-parallel slab is a piece of glass with two flat, parallel faces — a microscope slide, a mobile phone screen glass, a window pane.

The claim: after passing through the slab, the emergent ray is parallel to the incident ray. It is not in the same line — it has been shifted sideways by a distance d called the lateral displacement. But the direction is unchanged.

Why the emergent ray is parallel

At the top surface, n_1\sin\theta_1 = n_2\sin\theta_2 (medium 1 is air, medium 2 is glass). At the bottom surface, the ray goes from glass back into air, so

But both equations have n_2\sin\theta_2 on one side. Combining:

The emergent ray makes the same angle with the normal as the incident ray. Since the two surfaces are parallel, their normals are parallel too, and the emergent ray is parallel to the incident ray. The slab does not change direction — it only shifts.

Computing the lateral displacement

The shift d comes from simple triangle geometry. The slab has thickness t. Inside the slab, the ray travels a distance L = t/\cos\theta_2 along the slanted path. The sideways shift in that path, measured perpendicular to the incident ray direction, comes from the extra angle (\theta_1 - \theta_2) between the slanted path inside and what the path would have been if there were no refraction.

Step 1. Length of slanted path inside the slab:

Why: the ray enters at the top, exits at the bottom; vertical distance t; angle from vertical \theta_2. So t = L\cos\theta_2, hence L = t/\cos\theta_2.

Step 2. If there were no slab, the ray would have continued at angle \theta_1 from the normal and travelled the same vertical distance t in a slanted length t/\cos\theta_1. The two slanted paths (with-slab and without-slab) share the same starting point on the top surface and end at different points on the plane of the bottom surface — the difference between these endpoints, measured perpendicular to the exiting ray, is d.

Step 3. Drop a perpendicular from the actual exit point to the ghost ray's line. The perpendicular length is d. From the triangle with hypotenuse L (the actual path length in the slab) and angle (\theta_1 - \theta_2) at the entry point between the actual path and the ghost path:

Step 4. Substitute L = t/\cos\theta_2:

For small angles this reduces to d \approx t(\theta_1 - \theta_2), which for a slab of n = 1.5 at \theta_1 = 30° gives about t \cdot (30° - 19.47°) \cdot \pi/180 = 0.18t — an 18\% shift for a moderately oblique ray.

Apparent depth — why pools look shallow

A coin sits on the bottom of a 15-cm deep tray of water in a Pune home kitchen. You look down at it (nearly vertically). The coin seems to be only 11.3 cm below the surface. This is apparent depth.

The geometry uses small-angle Snell's law. For rays that are close to the normal (so \sin\theta \approx \tan\theta \approx \theta in radians), the law

becomes

Now consider a point source O at depth h below the water-air interface. A ray from O near the vertical reaches the surface at a horizontal distance x from the point directly above O. The angles are:

• Inside water: \theta_1 \approx \tan\theta_1 = x/h.
• In air after refraction: \theta_2 \approx \tan\theta_2 = x/h_{\text{apparent}}, where h_{\text{apparent}} is the depth the eye perceives — the depth at which the outgoing rays, traced backwards, appear to converge.

Snell (small angle) for water-to-air: n_{\text{water}} \cdot \theta_1 = n_{\text{air}} \cdot \theta_2, i.e. 1.33\cdot(x/h) = 1.00\cdot(x/h_{\text{apparent}}).

Cancelling x:

For h = 15 cm and n = 1.33, h_{\text{apparent}} = 11.28 cm. A coin in a Kanpur bathtub sits at 15 cm but fishes at 11.3 cm, and the bather's hand grabs water instead.

The reverse case — looking up from under water at an object in air — gives the opposite effect: objects above the surface appear farther than they really are, by a factor of n. This is why a fish that wants to lunge at a dragonfly hovering 20 cm above the surface must learn that the dragonfly is really only 20/1.33 = 15 cm above the water line. Fish evolve excellent refraction intuition.

Worked examples

Common confusions

• "Refraction means the light slows down, so it gets dimmer." No. Refraction is about the change of direction at the boundary; the intensity drops at each interface because part of the light is reflected (and a very small fraction absorbed), but that is a separate effect from the bending. The refracted ray carries almost all of the incident energy (for most angles well below the critical angle), and Snell's law governs only the direction.

• "A denser optical medium is the same as a denser substance." Usually yes, but not always. Water is less dense than diamond (no surprise), and its refractive index is also smaller (1.33 < 2.42). But glycerine (n = 1.47) is denser by mass than olive oil (n = 1.47) and has nearly the same n. The phrase "optically denser" means larger n, regardless of mass density. The word "denser" here is a legacy term — what matters is the speed of light in the medium.

• "Frequency changes when light enters a medium." No — the frequency stays the same. It is the wavelength that changes: \lambda_{\text{medium}} = \lambda_0/n. The frequency is set by the source (the oscillating atoms or the antenna) and does not change as the wave crosses a boundary. Colour, which your eye interprets from frequency, does not change when light goes from air into water — underwater objects look the same colour.

• "Snell's law only works for air-to-glass." It works for any two transparent media. Water-to-glass, oil-to-water, diamond-to-cubic-zirconia — every combination obeys n_1\sin\theta_1 = n_2\sin\theta_2. The indices are tabulated; once you know them, the law takes over.

• "If I draw the normal on the other side of the surface, I get a different answer." The normal is a perpendicular line — it has no "other side" for the purpose of measuring angles. Both \theta_1 and \theta_2 are measured from the same normal line, one in each medium. Drawing the normal once, and measuring both angles from it, is the correct and only approach.

• "A ray crossing from vacuum to air doesn't bend because both are the same." Air has n = 1.0003, not 1.0000. The bend is tiny but non-zero, and it is enough to matter for starlight reaching the top of the Indian Astronomical Observatory at Hanle, where a star near the horizon can appear displaced by over half a degree due to atmospheric refraction.

Where this leads next

• Total Internal Reflection — above a certain critical angle, light going from dense to rare refuses to escape. The physics of optical fibres, diamond sparkle, and mirage formation.
• Prism and Dispersion — combining refraction at two non-parallel faces produces the prism. Different wavelengths refract by different amounts, spreading white light into the seven colours of the rainbow.
• Thin Lenses — a lens is a slab with curved faces. The same Snell's law, applied at each surface, gives the lens formula 1/v - 1/u = 1/f.
• Reflection of Light — the partner law, valid at the same boundaries where refraction occurs. Every transparent interface reflects a little; the rest refracts.
• Spherical Mirrors — the curved-surface version of reflection, with a mirror formula structurally identical to the lens formula.