Fermat’s Principle & Geometrical Optics – Comprehensive Bullet Notes

Fermat’s Principle – Historical Context & Scope

• Pierre de Fermat (1601–1665) formulated the principle in 1658.
• Unifies all basic laws of geometrical optics: rectilinear propagation, reflection, refraction, reversibility of light, lens-maker’s formula, thin–lens formula, etc.
• Provides a single variational criterion from which ray paths are derived.

Optical Path Length (OPL)

• Geometric distance traversed: $S = v t$ (where $v$ is velocity, $t$ is time).
• Refractive index definition: $\mu = \dfrac{c}{v} \;\Rightarrow\; v = \dfrac{c}{\mu}$.
• Optical Path $\Big(OP\Big) = \mu S = c t$.
  • For fixed time, $\mu S$ is constant.
  • Equivalent air path: distance in vacuum that light would cover in the same time.
• Discrete media sequence: $OP = \sum<em>{i=1}^{n} \mu</em>i S_i$.
• Continuously varying medium (ray from P to Q): $OP = \int_P^Q \mu\,ds$.

Fermat’s Principle – Three Statements

• Least-Time Form (historical): “Of all possible paths between two points, a light ray chooses the one that minimises travel time.”
  • Mathematically: $\delta t = 0 \;\Leftrightarrow\; \delta !\int_P^Q \mu\,ds = 0$.
• Extrema/Stationary Form (modern): time (or OPL) is an extremum (minimum, maximum, or stationary).
  • Explains cases of stationary path (e.g., perfect imaging by a lens) or maximum path (certain reflections).
• Rectilinear Propagation Deduction: For a homogeneous isotropic medium $\mu = \text{const}$, so $\int \mu ds = \mu \int ds$ is extremised by the shortest geometrical distance → straight-line ray.

Derivation of Classical Laws via Fermat

Reflection at a Plane Mirror

• Setup: Incident point A to reflected point B via arbitrary point P on mirror.
• OPL: $L(x,z)=\sqrt{x^{2}+a^{2}+z^{2}}+\sqrt{(x-d)^{2}+b^{2}+z^{2}}$.
• Stationary condition $\tfrac{\partial L}{\partial z}=0\Rightarrow z=0$ → incident ray, normal and reflected ray co-planar (1st law).
• $\tfrac{\partial L}{\partial x}=0$ yields $\sin i = \sin r$ → angle of incidence equals angle of reflection (2nd law).

Refraction at a Plane Interface (Snell’s Law)

• Two media $\mu<em>1$ (incident) and $\mu</em>2$ (refracted).
• OPL through point K: $L=\mu<em>1\sqrt{x^{2}+a^{2}+z^{2}}+\mu</em>2\sqrt{(x-c)^{2}+b^{2}+z^{2}}$.
• Stationary wrt $z$ → co-planarity of incident, refracted rays & normal (1st law).
• Stationary wrt $x$ → $\mu<em>1\sin i = \mu</em>2\sin r$ (Snell).

Spherical Surfaces – Paraxial (Gaussian) Formulation

Refraction at a Single Spherical Surface

• Geometry: Object distance $x\,(\equiv -u)$, image distance $y\,(\equiv +v)$ measured from pole; radius $r\,(\equiv R).$
• Using small-angle (paraxial) approx & Snell: $\frac{\mu<em>2}{v}-\frac{\mu</em>1}{u}=\frac{\mu<em>2-\mu</em>1}{R}$.
• Applies universally with Cartesian sign convention.

Reflection at a Spherical Mirror

• Treat reflection as refraction with $\mu<em>1 = -\mu</em>2$.
• Mirror formula: $\frac{1}{u}+\frac{1}{v}=\frac{2}{R}$ or $\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$ where $f=\tfrac{R}{2}$.

Cartesian Sign Convention (New)

1. Rays incident from left.
2. Distances right of pole (along +X) positive; left negative.
3. Angles measured anticlockwise positive.
4. Heights above axis positive; below negative.
5. Curvature radius positive if centre lies to right of pole.

• Unifies equations (always use one formula and insert signs).

Thin Spherical Lens

Derivation

• Two refracting surfaces with radii $R<em>1, R</em>2$; lens index $\mu<em>2$; surrounding medium $\mu</em>1$.
• Applying single-surface formula twice and adding:
  $\frac{\mu<em>1}{v}-\frac{\mu</em>1}{u}=\Big(\mu<em>2-\mu</em>1\Big)\Big(\frac{1}{R<em>1}-\frac{1}{R</em>2}\Big).$
• Lens surrounded by same medium ($\mu_1=1$ in air) → thin-lens (Gaussian) formula: $\boxed{\frac{1}{v}-\frac{1}{u}=\frac{1}{f}}.$

Lens-Maker’s Formula

• For object at infinity $u=\infty,\,v=f$:
  $\frac{1}{f}=(\mu-1)\Big(\frac{1}{R<em>1}-\frac{1}{R</em>2}\Big).$
• Sign of $f$ decides converging (f>0) or diverging (f<0) behaviour; depends on indices & curvatures.

Principal Foci & Focal Lengths

• First principal focus $F<em>1$: object position producing emergent rays parallel to axis. • Distance $CF</em>1=f<em>1$, formula $\dfrac{1}{f</em>1}=(\mu-1)\Big(\dfrac{1}{R<em>1}-\dfrac{1}{R</em>2}\Big)$.
• Second principal focus $F<em>2$: image point for object at infinity. $\dfrac{1}{f</em>2}=(\mu-1)\Big(\dfrac{1}{R<em>1}-\dfrac{1}{R</em>2}\Big)$ → $f<em>1=-f</em>2$ (symmetry, opposite sign).
• Focal planes: planes ⟂ axis through $F<em>1$ and $F</em>2$.

Newton’s Lens Equation

• Measure object & image from respective focal points: $x<em>1x</em>2=f<em>1f</em>2$.
• For same medium both sides ($f<em>1=-f</em>2=f$): $x<em>1x</em>2=-f^2$ (opposite signs confirm object & image on opposite sides).

Lateral (Transverse) Magnification

• Definition: $m=\dfrac{h<em>2}{h</em>1}= -\dfrac{v}{u}= -\dfrac{x<em>2}{f</em>2}= -\dfrac{f<em>1}{x</em>1}$.
• m>0 → erect image, m<0 → inverted.
• For multi-element system: total magnification $m=m<em>1m</em>2…$

Silvered Lenses & Combination Systems

• Treat reflection at silvered surface as mirror with effective focal length $F$; derive via lens + mirror in series:
  $\frac{1}{F}=\frac{2}{R}+\frac{2(\mu-1)}{R}\;\text{(depending on which surface is silvered)}.$
• Special cases:
  • Plane-silvered side $f_m=\infty$ simplifies expressions.
  • Back-surface silvered plano-convex behaves as concave mirror $F=-\dfrac{R}{2(\mu-1)}$.

Ray-Tracing Rules (Paraxial)

1. Ray through $F_1$ → emerges parallel.
2. Ray parallel to axis → passes through (or appears from) $F_2$.
3. Ray through optic centre $C$ → undeviated.

Typical Numerical Illustrations

• Worked examples include determination of image position through glass sphere, combination of lens & mirror, two-element lens systems, etc.
• Emphasise consistent sign use.

Summary of Key Formulae

• Optical path (discrete): $OP=\sum \mu<em>i S</em>i$.
• Optical path (continuous): $OP=\int \mu\,ds$.
• Fermat stationary condition: $\delta\,OP=0$.
• Snell: $\mu<em>1\sin i = \mu</em>2\sin r$.
• Single spherical refraction: $\dfrac{\mu<em>2}{v}-\dfrac{\mu</em>1}{u}=\dfrac{\mu<em>2-\mu</em>1}{R}$.
• Spherical mirror: $\dfrac{1}{u}+\dfrac{1}{v}=\dfrac{2}{R}=\dfrac{1}{f}$.
• Thin lens (Gaussian): $\dfrac{1}{v}-\dfrac{1}{u}=\dfrac{1}{f}$.
• Lens-maker: $\dfrac{1}{f}=(\mu-1)\Big(\dfrac{1}{R<em>1}-\dfrac{1}{R</em>2}\Big)$.
• Newton: $x<em>1x</em>2=f<em>1f</em>2$.
• Magnification: $m=-\dfrac{v}{u}$.

Conceptual / Ethical / Practical Notes

• Variational ideas (least action/time) later became cornerstone for Hamiltonian optics and modern physics.
• Practical lens design (cameras, spectacles, telescopes) uses lens-maker relation to select curvatures.
• Understanding sign convention critical to avoid design errors.
• Optical path concept underlies interferometry and fibre-optic communication where phase equality is essential.

Connections & Further Studies

• Fermat’s principle ↔ Huygens’s principle ↔ Principle of least action (Euler-Lagrange).
• In GR, light follows null geodesics—a curved-space generalisation of Fermat.
• Wave-optical derivations (eikonal equation) reduce to geometrical optics in short-wavelength limit.

University Model-Question Themes

• Definitions: optical path, principal foci, magnification.
• Proof questions: derive laws via Fermat, derive Gaussian & Newton formulae.
• Problem practice: applying sign convention, multi-element systems, silvered lenses, refractive index calculations.