Foundations — Snell's law — derivation from Fermat's principle

Before you can read the parent derivation of Snell's law, every symbol it uses must first mean something you can see. This page builds each one from absolute zero, in the order they depend on each other.

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1. A point and a straight line (the stage)

Everything happens on a flat drawing. Two special points live on it:

• $A$ — where the light starts (up in the fast material).
• $B$ — where the light ends (down in the slow material).

Between them runs a flat horizontal line: the boundary (also called the interface or surface). Above it is material 1; below it is material 2.

We need $A$, $B$, and the boundary because the whole question is: "starting at $A$, ending at $B$, where should light cross the line?"

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2. Distance $d$, and speed $v$

The picture: two dots joined by a segment of length $d$; a little runner covers it in time $t$.

We need $v$ because the whole "least time" story is about light being fast in some materials and slow in others. Different speeds are the reason it bends at all.

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3. Time $t$, and the rule $t=d/v$

Why divide distance by speed? Because if you cover $v$ metres each second, then to cover $d$ metres you simply need $d/v$ seconds.

$t=\frac{d}{v}$

Look at the figure below. The dashed navy line is the shortest distance (a straight line from $A$ to $B$). The solid magenta line is the least time path: it crosses the boundary farther along so light spends more length in the fast top material and less in the slow bottom material. The kink at the boundary is exactly what Snell's law will pin down.

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4. The speed of light $c$, and refractive index $n$ (and $n_1$, $n_2$)

Inside glass or water, light is slower than $c$. We measure "how much slower" with one number:

Because Snell's law compares two materials, we tag the index with a number to say which material we mean:

The picture below: a bar labelled $c$ at the top (full speed), and a shorter bar $v=c/n$ below it — the bigger the $n$, the shorter the bar. Water's bar is shorter than air's; glass's is shorter still.

Why the topic needs $n$ (and both $n_1$, $n_2$): it lets us rewrite the awkward speed $v$ as $c/n$, so the time in medium 1 becomes $n_1{d}_1/c$ and in medium 2 becomes $n_2{d}_2/c$. See Refractive Index.

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5. Optical path length: $n\cdot d$

Since $t=nd/c$ and $c$ is a fixed constant, minimizing time is the same as minimizing the quantity $n\cdot d$. That quantity gets its own name:

Picture: a real ruler of length $d$, but the material stretches the ruler by a factor $n$ when you count the "cost". See Optical Path Length.

Why: it removes the constant $c$ from every calculation, so we compare paths using pure lengths and indices — no seconds needed.

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6. The geometry variables: $a$, $b$, $w$, $\ell$, and $x$

The derivation slices the trip into two straight legs and gives every measurement a name.

Look at the figure below to see all five at once: $a$ and $b$ are the vertical legs, $x$ and $w-x$ split the total width $w$, and each slanted magenta segment is an $\ell$.

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7. The right triangle and its sides

To measure angles we need a right triangle — a triangle with one square ($90^{\circ }$) corner.

Look at the figure below. Each leg of light forms exactly such a triangle: a vertical side ($a$ or $b$), a horizontal side ($x$ or $w-x$), and the slanted ray $\ell$ as the longest side.

The topic needs these names because the sine of an angle is built from them, and the whole derivation ends by reading these triangles.

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8. The normal, and the angle $\theta$

The picture: the flagpole (normal) stands up; the ray leans away from it by angle $\theta$. A ray hugging the pole has small $\theta$; a ray nearly flat along the water has $\theta$ near $90^{\circ }$.

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9. The sine, $\sin \theta$

Now that we have a triangle and an angle from the normal, we can define the tool that connects them.

In the derivation the run is $x$ (or $w-x$) and the hypotenuse is the path length $\ell$, so $\sin {\theta }_1=\frac{x}{\sqrt{a^2+x^2}}=\frac{\text{run}}{\text{ray length}}.$

The figure in Section 7 shows this ratio drawn on the triangle: the horizontal run $x$ over the slanted ray $\ell$.

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10. The square root, and $\sqrt{a^2+x^2}$

The picture: the ray from $A$ to the crossing point is the slanted hypotenuse of a right triangle with a vertical leg $a$ (the height of $A$ above the line) and a horizontal leg $x$ (how far sideways). Its length is $\sqrt{a^2+x^2}={\ell }_1$.

Why the topic needs it: it writes each straight path length $\ell$ in terms of the one thing light gets to choose — the crossing point $x$.

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11. The total time $T$

We now add up the two legs into a single quantity we can minimize.

The picture: a U-shaped curve of $T$ plotted against $x$; the true light path sits at the very bottom, where the curve is flat.

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12. Minimize, and the derivative $dT/dx$

Why this tool? Because "least time" means "bottom of the U", and the mathematical name for "the bottom, where the curve is flat" is the place where the derivative is zero. The derivative is the machine that finds that spot exactly, without guessing. This is the single calculus step the whole derivation rests on — and it acts on the total time $T$, not on a single leg.

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Prerequisite map

Note the two arrows into $T$ from both $c$ and the optical path length: $T=\text{OPL}/c$, so the constant $c$ is never dropped from the definition — only later, when both sides of $dT/dx=0$ share a $1/c$ that cancels.

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Equipment checklist

Test yourself — cover the right side and answer before revealing.

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Connections

• Snell's law — derivation from Fermat's principle
• Fermat's Principle
• Refractive Index
• Optical Path Length
• Calculus — minimization and stationary points
• Reflection — law from Fermat's principle
• Total Internal Reflection
• Critical Angle
• Huygens' Principle and Snell's Law