Dispersion of Light in a Prism

Aim a narrow beam of white light at a glass prism and watch it split into a rainbow: violet rays are deviated more than red ones because the refractive index of glass depends on the wavelength. Rotate the prism, change its angle and material — from water to diamond, find the angle of minimum deviation, catch total internal reflection and repeat Newton’s two famous experiments: recombine the spectrum into white light and prove that a prism no longer splits a pure colour.

The rays are computed by ray tracing: the law of refraction acts at every face, and the weak reflected rays follow the Fresnel formulas. The running flashes are light pulses: in glass they move slower (v = c/n), and the violet ones fall behind the red ones. Dashed lines are the face normals and the continuation of the incident ray; the yellow arc is the deviation angle δ.

What the screen shows (scale — angle from the central ray)

Refractive index of the glass n(λ)

Deviation angle δ versus angle of incidence i

Experiment

Light source, nm

White light (all colours at once)

λ laser wavelength 589

Colour passed by the slit 530

Prism material

A apex angle of the prism 60°

i angle of incidence 54.0°

Light pulses (v = c/n)

Normals and angles i, A, δ

Check yourself: predict the result

Model: n(λ) comes from Sellmeier formulas with catalogue coefficients (Schott glasses BK7, F2, SF11; water at 20 °C; diamond). White light is 40 wavelengths from 400 to 700 nm. Rays are traced in the plane of the drawing with the law of refraction and the Fresnel formulas; the screen is always perpendicular to the beam. The pulses show the travel time: their speed ∝ 1/n.

n_r (656 nm)

n_v (405 nm)

δ_r

deviation of the red ray

δ_v

deviation of the violet ray

Δδ = δ_v − δ_r

angular width of the spectrum — a measure of dispersion

For λ = 589 nm

δ_min

minimum deviation angle

i = (A + δ_min) / 2

angle of incidence for the symmetric path

v = c / n

speed of light in the prism

sin α₀ = 1 / n

critical angle of total internal reflection

V = (n_D − 1) / (n_F − n_C)

Abbe number: the smaller it is, the more strongly the glass splits light

In 1666 Isaac Newton darkened a room, made a small hole in the shutter and placed a glass prism in the path of a sunbeam. On the wall, instead of a round white spot, an elongated coloured band appeared — a spectrum: red, orange, yellow, green, cyan, blue, violet. Newton realised that the prism does not “colour” the light but separates it: white light is a mixture of all colours, and glass refracts each colour differently. The dependence of the refractive index on the wavelength is called dispersion.

KEY FORMULAS

Law of refraction

$$n_1\sin\alpha = n_2\sin\beta$$

α is the angle of incidence, β the angle of refraction (from the normal to the face); n₁, n₂ are the refractive indices of the media. Entering an optically denser medium, the ray bends towards the normal

▶ Detailed Explanation

Snell’s law follows from the fact that light travels at different speeds in different media: n = c/v. A wavefront falling obliquely on the boundary enters the glass with one side first and slows down there while the other side still runs in air — so the front turns. In a prism the law is applied twice: at the entrance face (air → glass) and at the exit face (glass → air).

Dispersion enters the law through n: each wavelength has its own n(λ). At an angle of incidence of 54° on flint glass, the red ray is refracted at 30.1° and the violet one at 29.4°. The difference is less than a degree, but after the second face it grows to several degrees.

Deviation of a ray by a prism

$$\delta = i_1 + i_2 - A, \qquad r_1 + r_2 = A$$

i₁ is the angle of incidence on the first face, i₂ the angle of emergence from the second, r₁ and r₂ the angles inside the glass, A the apex angle of the prism; δ is the angle between the incident and emergent rays

▶ Detailed Explanation

At the first face the ray turns by the angle i₁ − r₁, at the second by i₂ − r₂, both times towards the base. Adding the turns and using the fact that the normals to the faces meet at the angle A (hence r₁ + r₂ = A), we get δ = i₁ + i₂ − A. Each wavelength has its own r₁ = arcsin(sin i₁/n) and i₂ = arcsin(n·sin r₂), so each colour also has its own δ.

If n·sin r₂ > 1, the second angle does not exist: the ray undergoes total internal reflection at the second face. On the δ(i) graph this region is shaded red — at small angles of incidence light is “trapped” in the prism, violet earlier than red.

Refractive index from the minimum deviation

$$n = \frac{\sin\dfrac{A + \delta_{\min}}{2}}{\sin\dfrac{A}{2}}$$

δmin is the minimum deviation angle: it is reached when the ray passes through the prism symmetrically (i₁ = i₂, inside parallel to the base)

▶ Detailed Explanation

For the symmetric path r₁ = r₂ = A/2 and i₁ = i₂ = (A + δmin)/2. Substituting this into the law of refraction sin i₁ = n·sin r₁ gives the formula. This is how n is measured with high precision in laboratories: on a goniometer the prism is rotated until the spectral line stops and moves back, and the angle δmin is read.

For flint glass with A = 60° and the yellow sodium line 589 nm, δmin ≈ 48.2° at i ≈ 54.1°, hence n = sin 54.1° / sin 30° ≈ 1.620. In the simulator the “Find the minimum deviation” button smoothly rotates the prism into this position.

Thin prism: deviation and spectrum width

$$\delta \approx (n - 1)A, \qquad \Delta\delta \approx (n_{\text{ф}} - n_{\text{кр}})\,A$$

For a prism with a small angle A (and not too oblique incidence) the deviation is proportional to A and (n − 1); the angular width of the spectrum Δδ is proportional to the difference of the indices for violet and red light

▶ Detailed Explanation

For small angles the sines can be replaced by the angles themselves: i₁ ≈ n·r₁, i₂ ≈ n·r₂, and then δ = n(r₁ + r₂) − A = (n − 1)A. This shows that dispersion “works” through the difference in n: two colours are deviated by angles differing by (n_v − n_r)·A.

For flint glass n_v − n_r ≈ 0.036, so a thin prism with A = 10° gives a spectrum about 0.36° wide, and a prism with A = 60° about 3.6° (for a thick prism the width grows faster than this formula predicts). Dispersion is also why simple lenses give a coloured fringe — chromatic aberration; it is corrected by cementing lenses of crown and flint glass with different Abbe numbers.

Speed and wavelength of light in a medium

$$v = \frac{c}{n}, \qquad \lambda_{\text{ст}} = \frac{\lambda}{n}$$

c = 3·10⁸ m/s is the speed of light in a vacuum; in a medium light is n times slower, the frequency does not change at the boundary, so the wavelength also decreases n times

▶ Detailed Explanation

Dispersion means that light of different colours travels through glass at different speeds. In flint glass red light moves at about 1.856·10⁸ m/s and violet at about 1.816·10⁸ m/s. After 10 cm of glass the violet pulse lags behind the red one by about 12 picoseconds (1.2·10⁻¹¹ s) — tiny, but it is exactly this difference in speed that makes the rays refract differently.

Switch on the “Light pulses” in the simulator: the flashes run along the rays at a constant speed in air and slow down inside the prism (where they are closer together). Colour is set by frequency, not wavelength: in glass 532 nm green light has a wavelength of about 330 nm, but it stays green.

Critical angle of total internal reflection

$$\sin\alpha_0 = \frac{1}{n}$$

If light falls from inside on the glass–air boundary at an angle larger than α₀, it does not come out but is totally reflected; α₀ ≈ 41° for crown glass, ≈ 38° for flint, ≈ 24° for diamond

▶ Detailed Explanation

When light leaves glass for air, the angle of refraction is larger than the angle of incidence. At the angle of incidence α₀ the refracted ray grazes along the face (90°), and at a larger angle there is no refracted ray: sin β = n·sin α > 1. All the light is reflected back — hence the name “total”.

Since n depends on λ, the critical angle for violet light is slightly smaller than for red. So if you rotate the prism towards smaller angles of incidence, the violet edge disappears from the spectrum first — try it on the δ(i) graph. In diamond the critical angle is so small that a prism with A = 60° does not let light out through the second face at all.

At the boundary between air and glass light refracts according to the law n₁·sin α = n₂·sin β. The refractive index n shows how many times slower light travels in a substance than in a vacuum: v = c/n. For glass n depends slightly on the wavelength: in flint glass n ≈ 1.615 for red light (656 nm) and n ≈ 1.651 for violet (405 nm). The larger n, the more the ray turns at each face. So violet rays are deviated the most, red ones the least, and behind the prism the white beam spreads into a fan. The n(λ) graph shows that n grows towards the violet end of the spectrum — this is normal dispersion.

A prism turns the ray twice — at the entrance and at the exit face — and both times towards the base. If i₁ is the angle of incidence, i₂ the angle of emergence and A the apex angle of the prism, the total deviation is δ = i₁ + i₂ − A, and the angles inside the glass satisfy r₁ + r₂ = A. Rotate the prism with the i slider and watch the δ(i) graph: the deviation first decreases and then grows again. The minimum deviation occurs for a symmetric path, when the ray inside the prism runs parallel to the base. Press “Find the minimum deviation” — and the spectrum on the screen stops and turns back. The refractive index is measured from δmin: n = sin((A + δmin)/2) / sin(A/2).

Compare the materials. Water has n ≈ 1.33 and little dispersion — the spectrum is narrow, less than a degree. Crown glass gives a spectrum about 1.4° wide, flint about 3.6°, dense flint almost 10°: flint glasses contain lead oxide, which strongly increases both n and its dependence on λ. Optical designers describe dispersion by the Abbe number V: the smaller it is, the more strongly the glass splits light. The width of the spectrum also grows with the apex angle: for a thin prism δ ≈ (n − 1)·A, so at A = 30° the deviation is about half that at 60°.

If the angle of incidence on the second face from inside exceeds the critical angle (sin α₀ = 1/n), light does not leave through it — total internal reflection occurs. For diamond n ≈ 2.42 and α₀ ≈ 24°, so a diamond prism with A = 60° does not let light out through the second face at any rotation. Reduce A to 30° — the light comes out and splits into a bright spectrum. The cut of a brilliant is designed exactly this way: light is reflected many times inside the stone and leaves it in multicoloured flashes — the “fire”.

Repeat Newton’s experiments. In the “Two prisms” mode an identical prism stands inverted behind the first one: it deviates each colour by the same angle in the opposite direction, and a white beam comes out again — so white light really is made of colours. In the “crucial experiment” a board with a narrow slit lets through only one colour. The second prism deviates it, but it does not split into new colours: a pure (monochromatic) colour is not a mixture. Switch on the light pulses — inside the glass they move slower and closer together, and the violet ones fall behind the red ones: dispersion is precisely the different speed of light of different colours in matter.

Dispersion of Light in a Prism

KEY FORMULAS

Law of refraction

Deviation of a ray by a prism

Refractive index from the minimum deviation

Thin prism: deviation and spectrum width

Speed and wavelength of light in a medium

Critical angle of total internal reflection

More on this topic

Other simulators

Problems on this topic