3.1.1 Geometric Optics#

Prompts

What is Newton’s corpuscle theory of light? What does it predict for rectilinear propagation, reflection, and refraction in different media?
State Fermat’s principle. Why is the word stationary more precise than minimum?
Derive Snell’s law from the condition of stationary optical path length.
The optical path length \(L = \int n \, \mathrm{d}s\) plays the role of “action” for light. What is the analogous quantity in classical mechanics, and why is this parallel significant?
Geometric optics fails to explain certain experiments. What phenomenon is missing from the ray picture, and which theory accounts for it?

Lecture Notes#

Overview#

The nature of light was one of the central debates in the history of physics. Newton (1704) proposed the corpuscle theory: light consists of particles flying in straight lines. Huygens (1690) proposed the wave theory: light is a wave propagating through a medium. Both theories correctly derive the three geometric laws of optics, but they make different predictions for the deeper nature of light and fail for different phenomena.

This lesson develops geometric optics from the particle picture. The central result is Fermat’s principle of stationary optical path: a single variational principle from which all three laws of geometric optics follow—in the same way that Hamilton’s principle governs classical mechanics.

The Particle Theory of Light#

In Newton’s corpuscle model, light is a stream of particles. In a medium with refractive index \(n\), the speed of light is

(57)#\[ v = \frac{c}{n} \]

where \(c\) is the vacuum speed of light (\(n = 1\) in vacuum, \(n \approx 1.5\) in glass, \(n \approx 1.33\) in water).

From the particle picture, three empirical laws describe how light rays behave:

Law	Statement
Rectilinear propagation	In a uniform medium, light travels in straight lines.
Reflection	At a smooth mirror, angle of incidence equals angle of reflection.
Refraction	At an interface between media \(n_1\) and \(n_2\): \(n_1 \sin\theta_1 = n_2 \sin\theta_2\) (Snell’s law).

These laws describe mirrors, lenses, prisms, and optical fibers. All three follow from the variational principle below. The particle picture is powerful for geometric optics—but it provides no mechanism for diffraction (light bending around obstacles) or interference (two beams producing dark fringes). These phenomena prove decisive for the wave theory (§3.1.2).

Fermat’s Principle#

Fermat’s Principle

Light travels along the path for which the optical path length is stationary with respect to small variations:

(58)#\[ \delta L = 0, \quad L = \int_{\text{path}} n(\boldsymbol{r}) \, \mathrm{d}s \]

Here, \(n(\boldsymbol{r})\) is the refractive index and \(\mathrm{d}s\) is the arc length element along the path.

Why “stationary” and not “minimum”? In most cases the optical path length is a minimum for the true ray (Fermat’s least-time principle). However, at caustics and some curved interfaces, the path can be a maximum or saddle point. The key is that the first-order variation vanishes: \(\delta L = 0\).

Connection to mechanics: The optical path length \(L = \int n \, \mathrm{d}s\) is the action for light, just as \(S = \int L_{\rm mech} \, \mathrm{d}t\) is the mechanical action. Fermat’s principle (\(\delta L = 0\)) is the exact analog of Hamilton’s principle (\(\delta S = 0\)). This structural parallel becomes the bridge to quantum mechanics in §3.1.3 (Wave-Particle Duality).

Laws of Geometric Optics#

All three laws in the table above follow from Fermat’s principle.

Rectilinear Propagation.

In a uniform medium with constant \(n\), the path that makes \(L = \int n \, \mathrm{d}s\) stationary is simply a straight line.

Reflection.

When light reflects off a smooth boundary, the law of reflection follows:

\[ \theta_{r} = \theta_{i} \]

Derivation: Reflection from Fermat’s Principle

Consider a ray in a uniform medium reflecting off a flat mirror. By Fermat’s principle, the path minimizes the geometric path length (since \(n\) is constant).

By the geometry of reflection, “unfold” the mirror by reflecting the destination \(B\) to its mirror image \(B'\). The shortest path from source \(A\) to \(B'\) is a straight line. Folding back gives the actual ray path with angle of incidence = angle of reflection.

Refraction.

Snell’s Law: At an interface between media with refractive indices \(n_1\) and \(n_2\):

(59)#\[ n_1 \sin \theta_1 = n_2 \sin \theta_2 \]

Derivation: Snell’s Law from Fermat’s Principle

Place the interface at \(z = 0\), source \(A\) at \((0, 0, -h_1)\) and target \(B\) at \((d, 0, h_2)\). The ray crosses the interface at \((x, 0, 0)\). The total optical path length is

\[ L(x) = n_1 \sqrt{x^2 + h_1^2} + n_2 \sqrt{(d-x)^2 + h_2^2} \]

Applying Fermat’s condition \(\mathrm{d}L / \mathrm{d}x = 0\):

\[ n_1 \frac{x}{\sqrt{x^2 + h_1^2}} = n_2 \frac{d - x}{\sqrt{(d-x)^2 + h_2^2}} \]

Identifying \(\sin \theta_1 = x / \sqrt{x^2 + h_1^2}\) and \(\sin \theta_2 = (d-x) / \sqrt{(d-x)^2 + h_2^2}\) recovers Snell’s law.

Discussion: stationary need not mean shortest

Fermat’s principle says light takes the path of stationary optical path length, not necessarily the shortest. Can you construct a physical situation where the true ray path is a saddle point of \(L\) rather than a minimum? What does this tell us about the relationship between variational principles and physical behavior?

Poll: Fermat’s principle minimization

Fermat’s principle states that light travels along paths where the optical path length is stationary. In most cases, this path minimizes the optical path length. Which of the following correctly explains why “stationary” is more precise than “minimum”?

(A) The optical path length depends on wavelength, so the minimum varies with color.

(B) At certain geometries (e.g., caustics), the physical ray can correspond to a saddle point or maximum of the optical path length, not a minimum.

(D) Fermat’s principle only works when \(n\) is constant, in which case paths are always minima.

Summary#

Historical debate: Newton’s corpuscle theory and Huygens’ wave theory both predict the same geometric laws, but the corpuscle picture offers no mechanism for diffraction or interference. These phenomena ultimately reveal the wave nature of light (§3.1.2).
Speed of light in media: \(v = c/n\); the refractive index \(n > 1\) encodes how a medium slows light.
Fermat’s Principle: Light travels paths where the optical path length \(L = \int n \, \mathrm{d}s\) is stationary. All three laws of geometric optics follow from \(\delta L = 0\).
Action parallel: Optical path length plays the role of action for light; Fermat’s principle (\(\delta L = 0\)) mirrors Hamilton’s principle (\(\delta S = 0\)). This parallel is the key to the path integral (§3.1.3, Wave-Particle Duality).

Homework#

1. Reflection from Fermat. Use Fermat’s principle to derive the law of reflection. Consider a ray from \(A\) to \(B\) reflecting off a flat mirror. Show that the angle of incidence equals the angle of reflection.

2. Total internal reflection. Light travels from glass (\(n=1.5\)) into air (\(n=1.0\)).

(a) Find the critical angle \(\theta_c\) beyond which total internal reflection occurs.

(b) Explain physically, using Snell’s law, why no refracted ray exists for \(\theta > \theta_c\).

3. Stratified index gradient. A medium has refractive index \(n(z) = n_0(1 + \alpha z)\) for small \(\alpha > 0\). Argue qualitatively, using Fermat’s principle, that a horizontal light ray will curve downward. (Hint: which path — straight or curved — has a smaller optical path length?)

4. Stationary versus minimum. In the standard Snell’s law geometry, the source \(A\) is at \((0,-h_1)\), the target \(B\) at \((d,h_2)\), and the ray crosses the interface at \((x,0)\). The optical path length is \(L(x) = n_1\sqrt{x^2+h_1^2} + n_2\sqrt{(d-x)^2+h_2^2}\). Show that \(\mathrm{d}^2 L/\mathrm{d}x^2 > 0\), confirming the stationary path is a minimum, not a maximum.

5. Corpuscle versus wave speed. Newton’s corpuscle theory predicts that light speeds up when entering a denser medium (the particles are attracted inward). The wave theory predicts light slows down: \(v = c/n < c\) for \(n > 1\). The Foucault experiment (1850) measured the speed of light in water and found it less than \(c\).

(a) Which theory does this experimental result support?

(b) Snell’s law \(n_1 \sin\theta_1 = n_2 \sin\theta_2\) follows from Fermat’s principle regardless of whether light speeds up or slows down in the denser medium. Does Fermat’s principle depend on the direction of the speed change, or only on the value of \(n\)?

6. Curved mirror focusing. A concave spherical mirror of curvature radius \(R\) has its vertex at the origin and its axis along \(z\). A point light source sits on the axis at distance \(u\) from the vertex. A reflected ray leaves the source, hits the mirror at a point \(P\) that is height \(h\) above the axis, and returns to an image point on the axis at distance \(v\) from the vertex.

(a) In the paraxial regime (\(h \ll R\)), show that the mirror surface at height \(h\) is displaced from the vertex by approximately \(h^2/(2R)\) along the axis.

(b) Expand the total optical path length \(L(h) = d_1(h) + d_2(h)\) — where \(d_1\) is the source-to-\(P\) distance and \(d_2\) is the \(P\)-to-image distance — to second order in \(h\), and show that

\[ L(h) \approx (u + v) + \frac{h^2}{2}\!\left(\frac{1}{u} + \frac{1}{v} - \frac{2}{R}\right). \]

(c) Apply Fermat’s principle: require that \(L\) is stationary with respect to \(h\) for every paraxial ray. Derive the mirror equation \(1/u + 1/v = 2/R\).

7. Optics-mechanics dictionary. The optical path length \(L = \int n \,\mathrm{d}s\) plays the role of action for light, and Fermat’s principle (\(\delta L = 0\)) mirrors Hamilton’s principle (\(\delta S = 0\)).

(a) In a uniform medium (\(n = \text{const}\)), Fermat’s principle reduces to finding the shortest geometric path. What is the analogous statement for a free particle under Hamilton’s principle?

(b) In a medium with spatially varying \(n(\boldsymbol{r})\), light curves toward regions of higher \(n\) (cf. Problem 3). What mechanical system exhibits analogous behavior — a particle curving toward regions of lower potential energy — and what quantity plays the role of \(n\)?