Snell’s Law of Refraction and Total Internal Reflection

Turn the laser around the protractor and choose the media — air, water, glass, diamond. The simulator measures the angles of incidence and refraction, checks Snell’s law sin α / sin β = n₂/n₁, finds the critical angle for total internal reflection and uses the Fresnel equations to show how much light is reflected and how much is transmitted.

Upper medium (n₁)

n₁ refractive index

Lower medium (n₂)

n₂ refractive index

Laser

α angle of incidence °

Sine construction (sin α and sin β)

Wavefronts: λ = λ₀/n, v = c/n

Plane-parallel slab

Test yourself: predict the result

Drag the laser right along the protractor. Ray brightness follows the Fresnel equations for unpolarised light; the exact fractions R and T are labelled on the rays. To see total internal reflection, put the laser in the denser medium and increase the angle.

Try this: put the laser in the glass (the “in medium 2” button) and slowly tilt the beam. The refracted ray moves further and further from the normal, fades, and at 41.8° lies along the boundary. A little more and it disappears: all the light is reflected back, as if from a perfect mirror. This is total internal reflection. Replace the glass with diamond and the critical angle drops to 24.4° — that is why a cut diamond sparkles so much. And if you switch on the wavefronts, you will see why the ray turns at all: in glass the wave travels n times slower, and its front wheels round like a line of soldiers stepping from asphalt onto sand.

KEY FORMULAS

Law of refraction (Snell’s law)

$$\frac{\sin\alpha}{\sin\beta} = \frac{n_2}{n_1} \quad\Leftrightarrow\quad n_1\sin\alpha = n_2\sin\beta$$

α is the angle of incidence, β the angle of refraction (both measured from the normal to the boundary); n₁ is the refractive index of the medium the light comes from, n₂ of the medium it enters

▶ Detailed Explanation

The law follows from the fact that light travels at different speeds in different media. A wavefront hitting the boundary obliquely has one edge already in the second medium, slowed down, while the other edge still runs in the first — so the front turns (Huygens’ principle). In the same time the wave covers distances v₁t and v₂t, hence sin α / sin β = v₁/v₂ = n₂/n₁.

Example: a ray from air strikes glass (n = 1.5) at 40°. sin β = sin 40° / 1.5 = 0.643 / 1.5 = 0.429, β ≈ 25.4°. Conversely, from glass into air at 25.4° the ray leaves at 40° — light paths are reversible. The incident ray, the refracted ray and the normal lie in one plane.

Absolute refractive index

$$n = \frac{c}{v}, \qquad \lambda = \frac{\lambda_0}{n}$$

c = 3·10⁸ m/s is the speed of light in a vacuum, v the speed of light in the medium, λ₀ the wavelength in a vacuum, λ in the medium

▶ Detailed Explanation

The refractive index shows how many times slower light travels in a substance than in a vacuum: for air n ≈ 1.0003 (problems use 1), for water 1.33 (v ≈ 2.26·10⁸ m/s), for glass about 1.5 (v = 2·10⁸ m/s), for diamond 2.42 (v ≈ 1.24·10⁸ m/s). A medium with a larger n is called optically denser.

When light passes from one medium to another, its frequency ν does not change (as many crests leave the boundary as arrive), but the speed v = λν decreases — so the wavelength decreases too: λ = λ₀/n. Switch on the wavefronts in the simulator: in glass they are closer together and slower.

Critical angle of total internal reflection

$$\sin\alpha_0 = \frac{n_2}{n_1}, \qquad n_1 > n_2$$

Light goes from an optically denser medium (n₁) into a less dense one (n₂). For α > α₀ there is no refracted ray — all the light is reflected

▶ Detailed Explanation

From the law of refraction sin β = (n₁/n₂)·sin α. For n₁ > n₂ the right-hand side grows faster than sin α and reaches one at the angle α₀: β = 90°, the refracted ray grazes the boundary. For larger angles the equation has no solution — the light is totally reflected back. At a boundary with air: water — α₀ ≈ 48.8°, glass — 41.8°, diamond — 24.4°. For glass (1.5) against water (1.33): sin α₀ = 1.33/1.5 = 0.887, α₀ ≈ 62.5°.

The transition to TIR is smooth: as α approaches α₀ the refracted ray fades and the reflected one brightens (Fresnel equations). Total internal reflection underlies optical fibres, prism binoculars and periscopes, and reflectors; diamond’s small critical angle makes light bounce many times inside a cut stone, so it sparkles.

Fraction of reflected light (Fresnel)

$$R = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2, \qquad T = 1 - R$$

Reflectance R at normal incidence; T is the fraction of transmitted light. At oblique incidence R is given by the full Fresnel equations and grows with angle

▶ Detailed Explanation

At normal incidence glass (n = 1.5) reflects (0.5/2.5)² = 4 % of light, water about 2 %, diamond 17 %. For oblique incidence the two polarisations are treated separately: r_s = (n₁cos α − n₂cos β)/(n₁cos α + n₂cos β), r_p = (n₂cos α − n₁cos β)/(n₂cos α + n₁cos β), and for ordinary (unpolarised) light R = (r_s² + r_p²)/2. That is how ray brightness is computed in the simulator.

As the angle grows, R increases and tends to 100 % at grazing incidence — that is why a distant lake mirrors the sky. At Brewster’s angle tan α_B = n₂/n₁ the reflected light is fully polarised: the r_p component vanishes and only s-polarised light is reflected. Under total internal reflection R = 1.

Lateral shift of a ray by a plane-parallel slab

$$x = \frac{d\,\sin(\alpha - \beta)}{\cos\beta}$$

d is the slab thickness, α the angle of incidence, β the angle of refraction inside the slab; the ray leaves parallel to the incident one

▶ Detailed Explanation

Inside the slab the ray travels a path d/cos β. The continuation of the incident ray deviates from it by the angle α − β, so the distance between the incident and the emerging rays is x = (d/cos β)·sin(α − β). At the second face the angle of incidence is β, and by the law of refraction the ray leaves at α — parallel to the incident ray.

Example: a glass slab d = 4 cm, n = 1.5, α = 45°. sin β = 0.707/1.5 = 0.471, β ≈ 28.1°; x = 4·sin 16.9° / cos 28.1° ≈ 1.32 cm. At α = 0 there is no shift; at grazing incidence x tends to d. There is no total internal reflection in a slab: sin β < n₁/n₂.

The law of refraction. At the boundary between two media a light ray is partly reflected and partly transmitted into the other medium, changing direction. The incident ray, the refracted ray and the normal lie in one plane, and the angles obey Snell’s law: n₁·sin α = n₂·sin β, or sin α / sin β = n₂/n₁. The angle of incidence α and the angle of refraction β are measured from the normal — the perpendicular to the boundary — not from the surface itself. This is a common mistake in problems.

Refractive index. The absolute refractive index n = c/v shows how many times slower light travels in a substance than in a vacuum: 1.33 for water (v ≈ 2.26·10⁸ m/s), about 1.5 for glass, 2.42 for diamond. The frequency of light does not change at the boundary, so the wavelength decreases: λ = λ₀/n. Going from an optically less dense medium into a denser one, the ray bends towards the normal (β < α); going back, it bends away (β > α). At normal incidence (α = 0) the ray passes straight through.

Total internal reflection. If light goes from a denser medium into a less dense one (n₁ > n₂), the angle β is always larger than α. At the angle of incidence α₀ for which sin α₀ = n₂/n₁, the refracted ray grazes the boundary (β = 90°). For α > α₀ there is no refracted ray at all: the light is totally reflected. For water–air α₀ ≈ 48.8°, glass–air ≈ 41.8°, diamond–air ≈ 24.4°. Optical fibres, binocular and periscope prisms and reflectors all rely on this effect.

How much light is reflected. The fractions of reflected and transmitted energy are given by the Fresnel equations. At normal incidence R = ((n₁ − n₂)/(n₁ + n₂))² is reflected: about 4 % from glass and about 2 % from water. As the angle grows, R increases and tends to 100 % at grazing incidence — that is why distant water shines like a mirror. The transmitted fraction is T = 1 − R. As the critical angle approaches, the refracted ray fades out and the reflected one brightens — the transition to TIR is smooth. In the simulator the ray brightness is computed from these formulas, and the exact percentages are labelled on the rays and plotted on the R(α) graph.

Plane-parallel slab. After passing through a slab with parallel faces, the ray leaves parallel to the incident ray but shifted by x = d·sin(α − β)/cos β, where d is the slab thickness. There is no total reflection inside the slab: at the second face the angle of incidence is β, and sin β < n₁/n₂. The thicker the slab and the larger the angle, the larger the shift — that is how a window pane slightly displaces objects behind it.

Snell’s Law of Refraction and Total Internal Reflection

KEY FORMULAS

Law of refraction (Snell’s law)

Absolute refractive index

Critical angle of total internal reflection

Fraction of reflected light (Fresnel)

Lateral shift of a ray by a plane-parallel slab

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