Dispersion and Prisms

Hold a triangular prism in a shaft of sunlight and the white beam walks out as a ribbon of colour — red on top, violet on the bottom, every hue in between. Isaac Newton did exactly this in 1666 and drew the right conclusion: the prism does not paint the light, it sorts it. White light is already a mixture of colours, and the glass bends each by a slightly different angle. That differential bending is dispersion, and the prism makes it visible. The mechanism is Snell's law applied twice. A ray refracts at the first face, crosses the glass, and refracts again at the second, leaving deviated by a total angle $\delta$ . Because the index $n$ is not one fixed number but a slowly falling function of wavelength, $n(\lambda)$ , each colour has its own $\delta$ : violet, with the largest index, deviates most; red, with the smallest, least. Feed in white light and the beam fans into a spectrum. Buried in that geometry is a special ray. As you rotate the prism, the deviation of any colour falls to a shallow minimum, then rises again — at the bottom the ray passes symmetrically, entering and leaving at equal angles. Because the deviation is stationary there, measuring the apex angle $A$ and the minimum deviation $\delta_{\min}$ yields the index directly, $n=\sin[(A+\delta_{\min})/2]/\sin(A/2)$ , to four decimals with nothing but a protractor and a spectral lamp. This article owns the ray-level story: the deviation geometry, the minimum-deviation formula and its inversion, the thin-prism limit, angular dispersion, and the Cauchy law $n(\lambda)=A_c+B_c/\lambda^2$ . Why the electrons of glass make $n$ fall with wavelength is taken up in the companion article Normal and Anomalous Dispersion.

Key takeaways

Dispersion is the wavelength dependence of the index, $n=n(\lambda)$ ; in a transparent glass $n$ falls as $\lambda$ grows, so violet (large $n$ ) bends more than red.
A prism deviates a ray by $\delta=\theta_1+\theta_4-A$ (two Snell refractions minus the apex $A$ ); because $n$ varies with colour, white light fans into a spectrum.
Rotate the prism and $\delta$ passes through a minimum at the symmetric passage $\theta_1=\theta_4$ , $\theta_2=\theta_3=A/2$ , where $\delta$ is stationary — the basis of the prism refractometer.
At minimum deviation the index reads off directly: $n=\sin\!\big[(A+\delta_{\min})/2\big]/\sin(A/2)$ ; for $A=60^\circ$ , $\delta_{\min}=38.9^\circ$ this gives $n=1.520$ .
For a thin prism the geometry collapses to $\delta\approx(n-1)A$ , and the Cauchy law $n=A_c+B_c/\lambda^2$ captures normal dispersion across the visible.
Key idea: a prism converts the material property $\mathrm{d}n/\mathrm{d}\lambda$ into a measurable angular spread $\mathrm{d}\delta/\mathrm{d}\lambda$ , and at the minimum-deviation point its geometry simplifies so cleanly that $n$ can be read straight from two angles.

The core insight

A prism turns the invisible fact that $n$ depends on wavelength into a visible fan of angles. Each colour obeys the same law, $\delta=\theta_1+\theta_4-A$ , but with its own index. That deviation is stationary at the symmetric passage, where the index is fixed by the apex angle and the minimum deviation alone — making the prism both the first spectroscope and a precision refractometer.

Where this lives today

Every prism spectrometer descends from Newton's setup: a collimated beam, a prism at minimum deviation, a detector reading angle against wavelength. Because the reading is stationary at $\delta_{\min}$ , laboratory refractometers still quote glass indices to five decimals from apex-angle and deviation measurements alone, and prism spectrographs — spreading light continuously, without a grating's overlapping orders — remain in service. The same $\mathrm{d}n/\mathrm{d}\lambda$ that fans a spectrum sets a prism's resolving power $R=\lambda/\Delta\lambda = b\,|\mathrm{d}n/\mathrm{d}\lambda|$ , growing with the base length $b$ the beam fills. Dispersion management is now an engineering discipline: chromatic aberration in camera and microscope lenses is corrected by the very achromatic doublets Newton thought impossible, and prism pairs in ultrafast laser labs compress femtosecond pulses whose colours would otherwise travel at different speeds. The same physics paints the rainbow: raindrops act as tiny prisms with an internal reflection, spreading sunlight across the roughly $2^\circ$ between the red and violet deviation angles to hang an arc near $42^\circ$ from the antisolar point.

The physical picture

Picture a marching column crossing at an angle from firm pavement onto soft mud: the flank that reaches the mud first slows first, so the whole rank pivots — that pivot is refraction. Now suppose the mud drags on the tall marchers (short wavelengths) a touch more than the short ones: each height pivots by a different amount, and a rank that entered together leaves as a fan. That is dispersion — one differential in how much each colour is slowed, amplified by geometry into a spread of directions. Inside the glass the light drives the bound electrons, which re-radiate a wave slightly retarded relative to the driver. The nearer the light's frequency creeps toward a natural resonance — for ordinary glasses, an ultraviolet absorption band — the more the wave is slowed. Blue sits nearer that resonance than red, so it sees a larger index. The result is normal dispersion: $n$ rises toward the blue, $\mathrm{d}n/\mathrm{d}\lambda<0$ . (The full oscillator account, and the reversal across a resonance, belong to Normal and Anomalous Dispersion.) A single prism does two things. It deviates: every colour bends toward the base by an angle $\delta$ growing with $n$ . And it disperses: because $n$ differs between colours, so does $\delta$ — white light enters, an ordered spectrum leaves, violet swung farthest from the incoming line.

Figure

A prism sorts white light: each colour refracts twice with its own index n(λ), so violet (largest n) is deviated most and red least; the deviation δ is measured from the incoming direction

Where along a prism's rotation does a colour bend least? Not at grazing or normal incidence, but at a symmetric sweet spot in between. Swing the input angle $\theta_1$ from small to large and the deviation $\delta$ starts high, drops to a shallow floor, and climbs again. The floor is the minimum deviation $\delta_{\min}$ , reached when the ray runs symmetrically through the prism — in one side and out the other at equal angles, parallel to the base inside. That symmetry is why the prism is a precision instrument: at the bottom of the valley $\delta$ is flat, so a small misalignment barely moves the reading.

Figure

Total deviation δ against angle of incidence θ₁ for a 60° prism, n = 1.52. The U-curve bottoms out at the symmetric passage, δ_min ≈ 38.9° at θ₁ = 49.5° — the minimum-deviation point that fixes n

The blue U is the entire ray-level physics of the prism on one plot: the floor marks the symmetric ray; the steep walls mark the approach to grazing incidence (right) and the total-internal-reflection limit at the exit face (left), beyond which no ray emerges. Read the floor off the axis — $\delta_{\min}\approx 38.9^\circ$ — and you have measured the glass.

The core equations

Everything below follows from Snell's law at each of the prism's two faces plus the geometry. Take the surrounding medium to be air ( $n_{\text{air}}\approx 1$ ) and the glass index $n$ . At the entry face $\theta_1$ refracts to the internal angle $\theta_2$ ; at the exit face the internal angle $\theta_3$ refracts to the emergent angle $\theta_4$ :

\sin\theta_1 = n\sin\theta_2, \qquad n\sin\theta_3 = \sin\theta_4 \qquad (1)

where: $\theta_1$ — angle of incidence at the first face; $\theta_2$ — internal refraction angle there; $\theta_3$ — internal angle of incidence at the second face; $\theta_4$ — angle of emergence; $n$ — refractive index of the prism. The same law bends the ray twice, once entering and once leaving, and only $n$ carries the colour dependence. The two internal angles are locked together by the shape of the glass: because the ray, the two surface normals, and the apex form a quadrilateral, the internal angles add to the apex angle, and the total deviation is what is left after subtracting it:

\theta_2 + \theta_3 = A, \qquad \delta = \theta_1 + \theta_4 - A \qquad (2)

where: $A$ — apex (refracting) angle; $\delta$ — total deviation, the angle between incident and emergent rays. The deviation is just the sum of the two face-bends $(\theta_1-\theta_2)+(\theta_4-\theta_3)$ with the apex constraint folded in. As the input angle is varied, $\delta$ passes through a minimum at the symmetric configuration $\theta_1=\theta_4$ , where $\theta_2=\theta_3=A/2$ . Setting $\theta_2=A/2$ in Snell's law and $\theta_1=(A+\delta_{\min})/2$ gives the most useful equation in prism optics, the refractometer formula:

n = \frac{\sin\!\left(\dfrac{A+\delta_{\min}}{2}\right)}{\sin\!\left(\dfrac{A}{2}\right)} \qquad (3)

where: $A$ — apex angle; $\delta_{\min}$ — minimum deviation at the wavelength of interest; $n$ — index at that wavelength. Because $\delta_{\min}$ depends on $\lambda$ , so does $n$ : measure $\delta_{\min}$ at several spectral lines and you map out the whole dispersion curve. For a thin prism — small apex angle, near-normal incidence — the sines linearise and the deviation loses its dependence on the input angle, becoming a fixed property of the glass and the wedge:

\delta \approx (n-1)\,A \qquad (4)

where: $\delta$ and $A$ in radians; $n$ — refractive index. This is the small-angle skeleton of $(3)$ , and it is the form that reappears inside the thin-lens and achromat equations, where a lens is treated as a stack of infinitesimal prisms. Finally, the wavelength dependence itself is captured empirically across the visible transparency window by the Cauchy series, whose leading two terms suffice for most glasses:

n(\lambda) = A_c + \frac{B_c}{\lambda^2} + \cdots \qquad (5)

where: $A_c$ — the long-wavelength (baseline) index; $B_c$ — the dispersion coefficient (units of length $^2$ ), always positive in a normal-dispersion region so that $n$ falls as $\lambda$ grows; $\lambda$ — vacuum wavelength. Equation $(5)$ is the low-order tail of the physically grounded Sellmeier form, expanded far from the material's ultraviolet resonance.

Derivation ADeviation through a prism and the minimum-deviation formula

The goal is to earn, not assume, the refractometer formula $(3)$ — to show that the deviation really does bottom out at the symmetric passage, and that there the index is fixed by the apex angle and the minimum deviation alone. We treat $\delta$ as a function of the single input angle $\theta_1$ at fixed $A$ and $n$ , find where it is stationary, and evaluate the geometry there.

Step 1. Write the deviation as the sum of the bends at the two faces: $\delta=(\theta_1-\theta_2)+(\theta_4-\theta_3)$ . Using the apex constraint $\theta_2+\theta_3=A$ this collapses to $\delta=\theta_1+\theta_4-A$ . Why: the constraint removes the two internal angles in favour of the one thing visible from outside — the apex angle.

Step 2. Once $\theta_1$ is chosen, everything downstream is forced: $\theta_2=\arcsin(\sin\theta_1/n)$ , $\theta_3=A-\theta_2$ , $\theta_4=\arcsin(n\sin\theta_3)$ . So $\delta$ is a function of the single variable $\theta_1$ , stationary where $\mathrm{d}\delta/\mathrm{d}\theta_1 = 1 + \mathrm{d}\theta_4/\mathrm{d}\theta_1 = 0$ . Why: a minimum of a one-variable function lives where its derivative vanishes — the flat floor of the U-curve.

Step 3. Differentiating the Snell relations and $\theta_2+\theta_3=A$ and setting $\mathrm{d}\delta/\mathrm{d}\theta_1=0$ forces $\cos\theta_1\cos\theta_3=\cos\theta_4\cos\theta_2$ , solved by the symmetric ray $\theta_1=\theta_4$ , $\theta_2=\theta_3=A/2$ : the ray crosses the glass parallel to the base. Why: the physics has a left-right symmetry, so its stationary point is the symmetric ray — the one that looks the same run forwards or backwards.

Step 4. Evaluate at the minimum. Symmetry gives $\delta_{\min}=2\theta_1-A$ , so $\theta_1=(A+\delta_{\min})/2$ and $\theta_2=A/2$ ; feed both into $\sin\theta_1=n\sin\theta_2$ to get $n=\sin[(A+\delta_{\min})/2]/\sin(A/2)$ . Inverting, $\delta_{\min}=2\arcsin\!\big(n\sin(A/2)\big)-A$ , so the map runs both ways. Why: this makes the prism a metrology tool — two angle measurements deliver $n$ , and the flat minimum makes the reading insensitive to small alignment errors. *The three key results — the identity $\delta=\theta_1+\theta_4-A$ , the vanishing of $\mathrm{d}\delta/\mathrm{d}\theta_1$ at the symmetric ray ( $\theta_2=\theta_3=A/2$ ), and the round-trip $A=60^\circ,\ \delta_{\min}=38.9^\circ \leftrightarrow n=1.520$ — all follow directly from the steps above.*

Go deeper · Derivation B · 1 minAngular dispersion and the thin-prism limit

Derivation BAngular dispersion and the thin-prism limit

The goal is to connect the material fact $\mathrm{d}n/\mathrm{d}\lambda$ to the thing you see — the angular spread of the spectrum — and to show how the geometry collapses to $\delta\approx(n-1)A$ when the prism is thin. We differentiate the inverted minimum-deviation formula, chain to wavelength, then Taylor-expand for a small apex angle.

Step 1. From $\delta_{\min}=2\arcsin\!\big(n\sin(A/2)\big)-A$ , differentiate with respect to index: $\dfrac{\mathrm{d}\delta}{\mathrm{d}n}=\dfrac{2\sin(A/2)}{\sqrt{1-n^2\sin^2(A/2)}}$ . Why: this factor is the geometric lever turning a change in index into a change in exit angle — it grows with apex angle, so steep prisms disperse more.

Step 2. Chain to wavelength: $\dfrac{\mathrm{d}\delta}{\mathrm{d}\lambda}=\dfrac{\mathrm{d}\delta}{\mathrm{d}n}\,\dfrac{\mathrm{d}n}{\mathrm{d}\lambda}$ . The first factor is pure geometry (apex angle and working index); the second is pure material. Their product is the angular dispersion, the fan per unit wavelength. Why: it separates what the optician controls, the prism shape, from what the glassmaker controls, the dispersion of the material.

Step 3. Evaluate for the prism of the figure, $A=60^\circ$ , $n=1.52$ : $\mathrm{d}\delta/\mathrm{d}n = 2(0.5)/\sqrt{1-1.52^2(0.5)^2}=1.539\ \text{rad}$ per unit index, i.e. $88.2^\circ$ of swing per unit $n$ . Over the small index step between red and blue this predicts the spectral spread — cashed out in Worked example 2. Why: it turns a measured index difference into a measured angle, closing the loop between material and instrument.

Step 4. Now take the apex angle small. With $A\ll1$ , $\sin(A/2)\approx A/2$ and $\arcsin(nA/2)\approx nA/2$ , so $\delta_{\min}=2\arcsin\!\big(n\sin(A/2)\big)-A \to nA-A=(n-1)A$ . The dependence on input angle has vanished to first order. Why: a thin prism deviates every near-axial ray by the same $(n-1)A$ regardless of how it is held — which is exactly why a lens, built from infinitesimal prism wedges, has a single focal length. *The three results above — the closed form $\mathrm{d}\delta/\mathrm{d}n=2\sin(A/2)/\sqrt{1-n^2\sin^2(A/2)}$ , its numerical value $1.539\ \text{rad}$ at $A=60^\circ, n=1.52$ , and the leading-order limit $\delta_{\min}\to(n-1)A$ (less than $1\%$ error at $A=5^\circ$ ) — all follow from the derivation steps above.*

Go deeper · Derivation C · 2 minCauchy's law and why the index falls with wavelength

Derivation CCauchy's law and why the index falls with wavelength

The goal is to justify the empirical shape $n(\lambda)=A_c+B_c/\lambda^2$ used throughout and to see why it must have $B_c>0$ in the transparent visible — why red always ends up at the bottom of the spectrum. We take the single-resonance form and expand it far from the resonance, which for common glasses sits in the ultraviolet.

Step 1. Model the glass with one dominant ultraviolet resonance at $\lambda_0$ : $n^2-1=\dfrac{B\lambda^2}{\lambda^2-\lambda_0^2}$ . In the visible $\lambda\gg\lambda_0$ , so write it as $n^2-1=\dfrac{B}{1-(\lambda_0/\lambda)^2}$ . Why: the single-oscillator Sellmeier term is the simplest form capturing a transparent window on the long-wavelength side of an absorption band.

Step 2. Expand the geometric series in $(\lambda_0/\lambda)^2$ : $n^2-1 = B\big(1+\lambda_0^2/\lambda^2+\cdots\big)$ . Taking the square root and regrouping the leading two terms gives $n\approx A_c+B_c/\lambda^2$ , with $A_c,B_c$ built from $B$ and $\lambda_0$ . Why: truncating after $1/\lambda^2$ is Cauchy's law — good to a few parts in $10^4$ across the visible.

Step 3. Read off the sign. Because $\lambda_0^2>0$ and $B>0$ , the coefficient $B_c$ is positive, so $\mathrm{d}n/\mathrm{d}\lambda=-2B_c/\lambda^3<0$ : the index falls as wavelength grows. Blue (short $\lambda$ ) has the larger index and deviates most; red the least. This is normal dispersion, the universal behaviour of a transparent medium below its resonances. Why: the sign follows from the resonance lying to the blue of the working band — the deep reason is the electron-oscillator response, developed in Normal and Anomalous Dispersion.

Step 4. Calibrate on real glass. Fitting $(5)$ to crown-glass indices at the hydrogen C line ( $656.3\ \text{nm}$ , $n_C=1.514$ ) and F line ( $486.1\ \text{nm}$ , $n_F=1.522$ ) gives $A_c=1.5043$ , $B_c=0.00419\ \mu\text{m}^2$ ; the same fit predicts the sodium-D index ( $589.3\ \text{nm}$ ) as $n_D=1.516$ , matching standard crown glass (BK7, $n_D=1.5168$ ). Why: a two-point fit that correctly predicts a third line shows the $1/\lambda^2$ law is capturing real physics, not just interpolating. *The results above — the Cauchy coefficients $A_c=1.5043$ , $B_c=0.00419\ \mu\text{m}^2$ fitted from the C and F lines, the predicted $n_D=1.516$ , and the sign $\mathrm{d}n/\mathrm{d}\lambda<0$ following from $B_c>0$ — all emerge directly from the steps of this derivation.*

Limiting cases

Case	Result	Meaning
Normal dispersion (visible transparency)	$\mathrm{d}n/\mathrm{d}\lambda<0$	Index rises toward the blue; violet deviates more than red. The everyday behaviour of ordinary glass.
Thin prism ( $A\ll1$ , near-normal incidence)	$\delta\approx(n-1)A$	Deviation independent of input angle to first order — the wedge hiding inside every lens.
Minimum-deviation symmetry	$\theta_1=\theta_4,\ \theta_2=\theta_3=A/2$	Symmetric passage; $\delta$ stationary, enabling the exact index formula $(3)$ .
Apex angle $A\to 0$	$\delta\to 0$	A flat slab disperses nothing — a prism must have a wedge to work.
Near an absorption band	$\mathrm{d}n/\mathrm{d}\lambda>0$ (anomalous)	Across a resonance $n$ falls with frequency and red can bend more than blue, alongside strong absorption — see Normal and Anomalous Dispersion.

Common trap“More bending always means more colour-spreading, so a colour-free lens is impossible”

This is Newton's own mistake, and it is worth learning because he made it. Newton believed a material's dispersion (how much it spreads colours) was rigidly tied to its refraction (how much it bends light), and concluded the chromatic blur of a lens was incurable and a colour-free refracting telescope impossible — partly why he turned to reflectors. The premise is false. Bending and colour-spreading are set by two independent numbers: the index $n$ fixes the bending, while the dispersive power — the Abbe number $V=(n_D-1)/(n_F-n_C)$ — fixes the spreading. Crown glass has a high Abbe number ( $V\approx 64$ ); flint glass a low one ( $V\approx 36$ ). Cement a converging crown element to a diverging flint element and their chromatic spreads cancel while their bending powers do not — a nearly colour-free achromatic doublet. John Dollond patented exactly this in 1758, refuting Newton outright. The lesson: dispersion is not locked to refraction — their ratio is a material choice.

Worked examples

1 · Measuring the index of a glass prism (the real measurement)

A prism of apex angle $A=60^\circ$ is set on a spectrometer table and rotated until the sodium-lamp image reaches its minimum deviation, read as $\delta_{\min}=38.9^\circ$ . What is the refractive index of the glass?

Step 1. At minimum deviation the ray is symmetric, so formula $(3)$ applies directly: $n=\sin[(A+\delta_{\min})/2]/\sin(A/2)$ — exact, needing only the two measured angles.

Step 2. Insert the numbers: $(A+\delta_{\min})/2=(60^\circ+38.9^\circ)/2=49.45^\circ$ and $A/2=30^\circ$ , so $n=\sin 49.45^\circ/\sin 30^\circ = 0.7599/0.5 = \mathbf{1.520}$ .

Step 3. Check the inversion. Feeding $n=1.520$ , $A=60^\circ$ back through $\delta_{\min}=2\arcsin(n\sin 30^\circ)-60^\circ = 2\arcsin(0.760)-60^\circ = \mathbf{38.9^\circ}$ reproduces the measured deviation. The value $n=1.520$ is a typical crown-glass index — the glass plotted in the punchline figure.

Step 4. The measurement is forgiving: because the U-curve is flat at its floor, a $0.5^\circ$ error in $\delta_{\min}$ shifts $n$ by only about $0.004$ — stationarity turns a protractor into a four-figure index.

2 · The red-to-blue spread of a crown-glass prism (exam level)

The same $60^\circ$ crown-glass prism is used at minimum deviation, with $n_C=1.514$ at the red C line ( $656\ \text{nm}$ ) and $n_F=1.522$ at the blue-green F line ( $486\ \text{nm}$ ). Through what angle does the spectrum fan out between these colours?

Step 1. Each colour has its own minimum deviation, from $\delta_{\min}=2\arcsin(n\sin(A/2))-A$ with $A=60^\circ$ , $\sin(A/2)=0.5$ .

Step 2. Red: $\delta_{\min}(C)=2\arcsin(1.514\times 0.5)-60^\circ = 2\arcsin(0.757)-60^\circ = 2(49.20^\circ)-60^\circ = 38.40^\circ$ .

Step 3. Blue: $\delta_{\min}(F)=2\arcsin(1.522\times 0.5)-60^\circ = 2\arcsin(0.761)-60^\circ = 2(49.55^\circ)-60^\circ = 39.11^\circ$ . The spread is $\Delta\delta = 39.11^\circ-38.40^\circ = \mathbf{0.71^\circ}$ — a little over half a degree, easily fanned into a broad spectrum on a distant screen.

Step 4. Cross-check with angular dispersion: using $\mathrm{d}\delta/\mathrm{d}n=1.539\ \text{rad}$ from Derivation 2, the index step $\Delta n=0.008$ predicts $\Delta\delta\approx 1.539\times 0.008 = 0.0123\ \text{rad}=\mathbf{0.71^\circ}$ — in exact agreement, confirming the fan width is a geometric lever times a material index step.

Physics of the portrait

The prism is where the study of colour became quantitative. Before Newton, colour was thought to be added to white light by the glass; he showed instead that white light already contains all the colours and the prism only separates what is there. He also, revealingly, got the next step wrong, and the honest lesson of his error outlived it.

Origins · 1 minIsaac Newton and how this was discovered

Isaac Newton

1643–1727 · Cambridge and London

In 1666, at Woolsthorpe during the plague years, Isaac Newton cast a prism's spectrum on the wall — then performed the decisive test he called the experimentum crucis: he isolated a single colour from the first prism and passed it through a second, finding it unchanged, neither further split nor altered in hue. This proved the colours were not manufactured by the glass but sorted from a mixture already in white light. He reported it in his 1672 letter "New Theory about Light and Colours" and expanded it into the Opticks of 1704. Yet from the same experiments Newton drew a firmly wrong conclusion: convinced that a material's dispersion was strictly proportional to its refraction, he declared an achromatic refractor impossible — disproved in 1758 when John Dollond built an achromatic doublet from crown and flint glass of different dispersive powers. Newton's genius established that white light is composite; his error, honestly recorded, taught opticians that bending and colour-spreading are separate knobs.

History

1637

— Descartes, in Les Météores, explains the rainbow geometrically as refraction and internal reflection in raindrops.

1666

— Newton's prism experiments at Woolsthorpe separate white light into an ordered spectrum.

1672

— Newton's letter "New Theory about Light and Colours" is read to the Royal Society.

1704

— Newton publishes Opticks, his synthesis of colour, refraction, and the spectrum.

1758

— John Dollond patents the achromatic doublet, refuting Newton's claim that colour-free refraction is impossible.

Going further

Texts. Hecht, Optics, 5th ed., ch. 5–6 (prisms, minimum deviation, Cauchy and Sellmeier, Abbe number) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., ch. 5 · Born & Wolf, Principles of Optics, ch. 4 · Demtröder, Laser Spectroscopy, Vol. 1 (prism spectrometers, resolving power). For the electron-oscillator origin of $n(\lambda)$ : Foot, Atomic Physics, ch. 2, and Loudon, The Quantum Theory of Light, ch. 2. The key numerical results in this article are: (1) $\delta=\theta_1+\theta_4-A$ from two Snell refractions and $\theta_2+\theta_3=A$ ; (2) $\mathrm{d}\delta/\mathrm{d}\theta_1=0$ at the symmetric ray $\theta_2=\theta_3=A/2$ , giving formula $(3)$ ; (3) the round trip $A=60^\circ,\ \delta_{\min}=38.9^\circ \leftrightarrow n=1.520$ ; (4) the crown-prism spread $\delta_{\min}(C)=38.40^\circ$ , $\delta_{\min}(F)=39.11^\circ$ , $\Delta\delta=0.71^\circ$ , consistent with $\mathrm{d}\delta/\mathrm{d}n=1.539\ \text{rad}$ ; (5) the Cauchy fit $A_c=1.5043$ , $B_c=0.00419\ \mu\text{m}^2$ and its prediction $n_D=1.516$ . The Lorentz-oscillator physics behind $\mathrm{d}n/\mathrm{d}\lambda$ is developed in Normal and Anomalous Dispersion.

Prerequisites. Snell's Law of Refraction · Refractive Index and Optical Path Length · Ray Model of Light · Total Internal Reflection and Critical Angle. Leads to. Thin Lens Equation and Lensmaker's Equation · Refraction at a Spherical Surface.

Is minimum deviation the same as zero deviation?

No. Zero deviation would mean the beam leaves parallel to how it entered, which a prism with a real apex angle never does. Minimum deviation is the smallest deviation as you rotate the prism, reached at the symmetric passage where the ray runs parallel to the base. There $\delta$ is stationary, which is why the reading is stable — but the deviation itself is substantial, about $38.9^\circ$ for the $60^\circ$ prism here, and differs for every colour.

Why does aligning to minimum deviation give such an accurate index?

Because the deviation is stationary there. Away from the minimum a small error in the prism angle changes $\delta$ at first order, corrupting the reading; at the minimum $\mathrm{d}\delta/\mathrm{d}\theta_1=0$ , so the same error changes $\delta$ only at second order — negligibly. The flat floor of the U-curve is an error-suppressing feature of the geometry, which is why a hand-rotated prism can deliver an index good to four or five decimals.

Why does violet bend more than red — what is physically different about it?

Inside the glass the light shakes the bound electrons, which have natural resonances in the ultraviolet. Violet, higher in frequency, sits closer to those resonances than red, so it drives the electrons harder, is slowed more, and sees a larger index — and a larger index means a larger deviation. Red bends least. The quantitative version is the Lorentz-oscillator model, worked out in Normal and Anomalous Dispersion.

What is the difference between a prism and a diffraction grating for making a spectrum?

Both spread light by wavelength, by different physics. A prism uses refraction: the spread follows $\mathrm{d}n/\mathrm{d}\lambda$ , is continuous and nonlinear (bunched toward the blue), with no overlapping orders. A grating uses interference: its spread is set by groove spacing, more nearly linear in wavelength, but with overlapping orders that must be filtered. Prisms are prized for throughput and order-free simplicity; gratings for higher, more uniform resolving power.

Does the apex angle have to be 60°?

No — $60^\circ$ is just the common equilateral choice, convenient because it puts typical glasses near a comfortable minimum-deviation angle. Any apex angle works; the formulas carry $A$ explicitly. There is an upper limit: if $A$ is too large the internal angle at the exit face exceeds the critical angle and the ray totally internally reflects instead of emerging — which is exactly how a $45^\circ$ – $90^\circ$ – $45^\circ$ prism is used as a perfect mirror rather than a disperser.

Why can a thin prism deviate light by the same amount regardless of how it is tilted?

Because in the thin-prism limit the deviation loses its dependence on the input angle: for a small apex angle and near-normal incidence, the two sines linearise and everything collapses to $\delta\approx(n-1)A$ — a constant fixed only by the index and the wedge angle. This makes the thin prism the right mental model for a lens: picture a lens as a stack of thin prisms of increasing wedge angle, each deviating its ray by $(n-1)A$ , all steering toward a common focus.