10.08.04 · em-sr / advanced-electrodynamics

Geometrical Optics as the Short-Wavelength Limit: Eikonal Equation and Ray Tracing

shipped3 tiersLean: none

Anchor (Master): Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 3; Born & Wolf, Principles of Optics, 7th ed. (1999), Ch. 3-4

Intuition Beginner

When the wavelength of light is far smaller than the objects and openings it encounters, light behaves as though it travels in straight lines. These straight-line paths are called rays, and the body of theory that uses them is geometrical optics (also called ray optics).

The wavelength of visible light is roughly 400 to 700 nanometres. A typical lens is centimetres across, a doorway is about a metre wide, and a room is several metres across. In all these cases the wavelength is millions of times smaller than the relevant structure. Light passing through a doorway casts a sharp shadow, not a diffraction pattern. Light entering the eye is focused by the cornea and lens onto the retina with pinpoint accuracy. These are geometrical-optics phenomena.

Three rules suffice for most ray-tracing problems:

Rectilinear propagation. In a uniform medium, rays are straight lines.
Law of reflection. At a mirror surface, the angle of incidence equals the angle of reflection, both measured from the surface normal.
Snell's law of refraction. When a ray crosses a boundary from a medium of refractive index $n_{1}$ into one of index $n_{2}$ :

n_{1} sin θ_{1} = n_{2} sin θ_{2}

A ray entering a denser medium ( $n_{2} > n_{1}$ ) bends toward the normal. A ray entering a less dense medium bends away from the normal.

A lens uses refraction at curved surfaces to redirect rays. A converging (convex) lens causes parallel incoming rays to meet at the focal point, a distance $f$ (the focal length) behind the lens. A diverging (concave) lens causes parallel rays to spread apart as though they originated from a virtual focal point in front of the lens. Given the object distance $d_{o}$ and image distance $d_{i}$ , the thin lens equation holds:

\frac{1}{d _{o}} + \frac{1}{d _{i}} = \frac{1}{f}

When $d_{o} > f$ , a real inverted image forms at distance $d_{i}$ . When $d_{o} < f$ , the lens acts as a magnifying glass and produces a virtual upright image.

Geometrical optics breaks down when the wavelength is no longer negligible compared to the obstacles or apertures. Light then diffracts 10.08.03: it bends around edges, spreads through narrow slits, and produces interference fringes. A shadow cast by light through a pinhole does not have a sharp boundary; instead the intensity tapers off in a pattern governed by the wave equation. The boundary between ray optics and wave optics is set by the ratio of wavelength to aperture size. When this ratio is small, geometrical optics is an excellent approximation. When it is of order unity, the full wave treatment is required.

This unit develops the mathematical framework that makes these statements precise. The central object is the eikonal equation, which emerges from the wave equation in the limit of vanishing wavelength and encodes the rule that rays travel along the direction of steepest increase in optical path length.

Visual Beginner

Regime	Condition	Behaviour	Examples
Ray optics	$λ$ much smaller than objects	Straight rays, sharp shadows	Cameras, eyeglasses, telescopes
Wave optics	$λ$ comparable to objects	Diffraction, interference, fringes	Pinholes, gratings, holograms

Optical element	Effect on rays	Governing rule
Flat mirror	Reflects at equal angle	$θ_{r} = θ_{i}$
Refracting surface	Bends ray per Snell's law	$n_{1} sin θ_{1} = n_{2} sin θ_{2}$
Converging lens	Brings parallel rays to focal point	$1/ d_{o} + 1/ d_{i} = 1/ f$
Curved mirror	Focuses or defocuses like a lens	$f = R /2$

Worked example Beginner

Example 1: Refraction through a glass prism.

A beam of monochromatic light in air ( $n_{1} = 1.00$ ) enters one face of a glass prism ( $n_{2} = 1.52$ ) at an angle of incidence $θ_{1} = 45°$ measured from the surface normal.

From Snell's law:

sin θ_{2} = \frac{n _{1}}{n _{2}} sin θ_{1} = \frac{1.00}{1.52} sin 45° = \frac{0.707}{1.52} = 0.465

θ_{2} = 27.7°

The ray bends toward the normal as it enters the denser glass. When the ray exits through the second face, Snell's law is applied again (now from glass to air), and the ray bends away from the normal. The total angular deviation depends on the apex angle of the prism and the refractive index. A prism disperses white light into its constituent colours because the refractive index depends on wavelength: blue light has a higher $n$ than red light and bends more, separating the spectrum.

Example 2: Total internal reflection and optical fibres.

A ray inside a glass fibre ( $n_{1} = 1.50$ ) approaches the glass-air boundary ( $n_{2} = 1.00$ ) at angle $θ_{1}$ from the normal. Snell's law gives:

sin θ_{2} = \frac{n _{1}}{n _{2}} sin θ_{1} = 1.50 sin θ_{1}

Since $sin θ_{2}$ cannot exceed 1, there is a critical angle $θ_{c}$ beyond which no refracted ray exists:

sin θ_{c} = \frac{n _{2}}{n _{1}} = \frac{1.00}{1.50} = 0.667, θ_{c} = 41.8°

For $θ_{1} > θ_{c}$ the ray is entirely reflected back into the glass. This is total internal reflection. An optical fibre confines light through repeated total internal reflections along its length, carrying signals over kilometres with minimal loss. The numerical aperture of the fibre determines the maximum entrance angle for which light remains trapped.

Example 3: The thin lens equation.

An object is placed 30 cm in front of a converging lens of focal length 10 cm. Using the thin lens equation:

\frac{1}{d _{o}} + \frac{1}{d _{i}} = \frac{1}{f} ⟹ \frac{1}{30} + \frac{1}{d _{i}} = \frac{1}{10}

\frac{1}{d _{i}} = \frac{1}{10} - \frac{1}{30} = \frac{3 - 1}{30} = \frac{2}{30} = \frac{1}{15}

d_{i} = 15 cm

A real image forms 15 cm behind the lens. The magnification is $M = - d_{i} / d_{o} = - 15/30 = - 0.5$ , so the image is inverted and half the size of the object.

Check your understanding Beginner

Exercise (easy, multiple choice).

Geometrical optics provides a good description of light when:

A. The wavelength is much larger than the objects and apertures involved. B. The wavelength is comparable to the objects and apertures involved. C. The wavelength is much smaller than the objects and apertures involved. D. The wavelength equals the refractive index times the object size.

Hint

Geometrical optics is the short-wavelength limit. It works when the wave nature of light produces effects too small to notice.

Answer

C. Geometrical optics assumes the wavelength $λ$ is much smaller than any characteristic length scale of the problem (aperture size, lens diameter, object size). In this limit, diffraction effects become negligible and light travels in straight rays. When $λ$ is comparable to these scales, diffraction 10.08.03 dominates and the ray picture fails.

Formal definition Intermediate+

The scalar wave equation in an inhomogeneous medium with position-dependent refractive index $n (r)$ is:

\nabla^{2} ψ - \frac{n ^{2} ( r )}{c ^{2}} \frac{\partial ^{2} ψ}{\partial t ^{2}} = 0

For a monochromatic wave $ψ (r, t) = u (r) e^{- iω t}$ , this reduces to the Helmholtz equation:

(\nabla^{2} + k^{2} n^{2} (r)) u = 0

where $k = ω / c$ is the free-space wavenumber.

Eikonal ansatz. Write the solution as:

u (r) = A (r) e^{i k_{0} S (r)}

where $k_{0} = 2 π / λ_{0} = ω / c$ is the free-space wavenumber, $A (r)$ is the slowly varying amplitude, and $S (r)$ is the eikonal (the optical path length function). The phase $k_{0} S$ varies rapidly in the short-wavelength limit $k_{0} \to \infty$ , while $A$ and $S$ vary on the scale of $n (r)$ , which is assumed large compared to $λ_{0}$ .

Eikonal equation. Substituting the ansatz into the Helmholtz equation and collecting the leading-order terms in $k_{0}^{2}$ yields:

∣\nabla S ∣^{2} = n^{2} (r)

This is the eikonal equation. Surfaces of constant $S$ are wavefronts. Rays are curves perpendicular to the wavefronts, directed along $\nabla S$ .

Transport equation. The next-order terms (linear in $k_{0}$ ) give energy conservation along a ray:

2\nabla A \cdot \nabla S + A \nabla^{2} S = 0

This is the transport equation for the amplitude $A$ . It ensures that the energy flowing through a ray tube is conserved.

Ray equation. Differentiating the eikonal equation along a ray curve $r (s)$ parametrised by arc length $s$ gives:

\frac{d}{d s} (n (r) \frac{d r}{d s}) = \nabla n (r)

This is the ray equation. In a uniform medium ( $n = const$ , $\nabla n = 0$ ), it gives $d^{2} r / d s^{2} = 0$ : straight-line rays. In a graded-index medium, rays curve toward regions of higher refractive index.

Fermat's principle. The ray equation is the Euler-Lagrange equation for the variational principle:

δ \int_{A}^{B} n (r) d s = 0

Light travels along paths of stationary optical path length $\int n d s$ . This is Fermat's principle (sometimes called the principle of least time, since $n = c / v$ makes the optical path length proportional to the travel time).

Paraxial approximation. When all rays make small angles with a preferred axis (say $z$ ), the arc length $d s \approx d z$ and the ray equation simplifies to:

\frac{d}{d z} (n \frac{d ρ}{d z}) = \frac{\partial n}{\partial ρ}

where $ρ = (x, y)$ is the transverse coordinate. This linearised equation is the workhorse of laser cavity design and fibre optics.

ABCD (ray-transfer) matrices. In the paraxial regime, a ray at height $y$ with angle $θ$ (small) relative to the axis is described by the column vector $(y θ)$ . Each optical element acts on this vector by a $2 \times 2$ matrix:

Element	ABCD matrix
Free-space propagation, distance $d$	$(10 d 1)$
Thin lens, focal length $f$	$(1 - 1/ f 01)$
Flat interface, $n_{1} \to n_{2}$	$(10 0 n_{1} / n_{2})$
Curved mirror, radius $R$	$(1 - 2/ R 01)$

A compound system of $N$ elements has an overall ABCD matrix equal to the product $M = M_{N} \dots M_{2} M_{1}$ (matrices multiplied in reverse order of traversal). The determinant of each matrix satisfies $det M = n_{1} / n_{2}$ , which equals 1 in a uniform medium.

Key derivation Intermediate+

Derivation (The eikonal equation from the Helmholtz equation).

Theorem. In the limit $k_{0} \to \infty$ , the Helmholtz equation $(\nabla^{2} + k_{0}^{2} n^{2}) u = 0$ reduces to the eikonal equation $∣\nabla S ∣^{2} = n^{2}$ , where $S$ is defined by the ansatz $u = A e^{i k_{0} S}$ .

Proof. Compute the Laplacian of $u = A e^{i k_{0} S}$ :

\nabla u = (\nabla A + i k_{0} A \nabla S) e^{i k_{0} S}

\nabla^{2} u = [\nabla^{2} A + i k_{0} (2\nabla A \cdot \nabla S + A \nabla^{2} S) - k_{0}^{2} A ∣\nabla S ∣^{2}] e^{i k_{0} S}

Substitute into the Helmholtz equation and divide through by $A e^{i k_{0} S}$ :

\frac{\nabla ^{2} A}{A} + i k_{0} (\frac{2\nabla A \cdot \nabla S}{A} + \nabla^{2} S) - k_{0}^{2} ∣\nabla S ∣^{2} + k_{0}^{2} n^{2} = 0

Separate real and imaginary parts:

Real part: $\frac{\nabla ^{2} A}{A} - k_{0}^{2} ∣\nabla S ∣^{2} + k_{0}^{2} n^{2} = 0$
Imaginary part: $\frac{2\nabla A \cdot \nabla S}{A} + \nabla^{2} S = 0$

In the short-wavelength limit $k_{0} \to \infty$ , the $k_{0}^{2}$ terms dominate in the real part. The term $\nabla^{2} A / A$ is of order 1 (assuming $A$ varies on the scale of $n$ ), while $k_{0}^{2} (∣\nabla S ∣^{2} - n^{2})$ is of order $k_{0}^{2}$ . Setting the leading-order coefficient of $k_{0}^{2}$ to zero:

∣\nabla S ∣^{2} = n^{2}

This is the eikonal equation. The imaginary part gives the transport equation at order $k_{0}$ :

2\nabla A \cdot \nabla S + A \nabla^{2} S = 0

which determines the amplitude once $S$ is known. The transport equation expresses energy conservation along a ray tube: as a bundle of rays converges ( $\nabla^{2} S > 0$ ), the amplitude must decrease to conserve flux; as it diverges ( $\nabla^{2} S < 0$ ), the amplitude increases. $□$

Derivation (The ray equation from the eikonal equation).

Theorem. The characteristic curves of the eikonal equation $∣\nabla S ∣^{2} = n^{2}$ satisfy the ray equation $\frac{d}{d s} (n \frac{d r}{d s}) = \nabla n$ .

Proof. Define the ray direction as $\hat{t} = \nabla S /∣\nabla S ∣ = \nabla S / n$ (unit vector along the ray). Parametrise the ray by arc length $s$ so that $d r / d s = \hat{t} = \nabla S / n$ .

Differentiate $n d r / d s = \nabla S$ with respect to $s$ :

\frac{d}{d s} (n \frac{d r}{d s}) = \frac{d}{d s} (\nabla S)

The right side is the directional derivative of $\nabla S$ along the ray:

\frac{d}{d s} (\nabla S) = (\frac{d r}{d s} \cdot \nabla) \nabla S = \frac{1}{n} (\nabla S \cdot \nabla) \nabla S

Using the eikonal equation $∣\nabla S ∣^{2} = n^{2}$ , take the gradient of both sides:

2 (\nabla S \cdot \nabla) \nabla S = \nabla (n^{2}) = 2 n \nabla n

Therefore $(\nabla S \cdot \nabla) \nabla S = n \nabla n$ , and:

\frac{d}{d s} (n \frac{d r}{d s}) = \frac{1}{n} \cdot n \nabla n = \nabla n

This is the ray equation. In a uniform medium $\nabla n = 0$ , giving $d (n \hat{t}) / d s = 0$ , so the ray travels in a straight line. In a graded medium, the ray curves toward regions of higher $n$ . $□$

Bridge. The eikonal equation is a first-order nonlinear PDE of Hamilton-Jacobi type. Its characteristics are the rays, which satisfy the ray equation. The foundational insight is that geometrical optics is the leading-order term in an asymptotic expansion of the wave equation in inverse powers of the wavenumber $k_{0}$ ; higher-order corrections capture diffractive effects 10.08.03. The central message is that the eikonal $S$ encodes the phase (wavefront geometry), while the amplitude $A$ is determined by energy conservation along rays via the transport equation. Putting these together, Fermat's principle emerges as the variational formulation of the ray equation, and Snell's law, the lens maker's equation, and the ABCD matrix formalism all follow as consequences. The connection to Hamilton-Jacobi theory in classical mechanics 09.04.09 reveals that ray optics and Newtonian mechanics share the same mathematical structure -- a correspondence that was the historical seed of wave mechanics.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Derive Snell's law from Fermat's principle. A ray travels from point $A = (0, - a)$ in medium 1 ( $n_{1}$ ) to point $B = (d, b)$ in medium 2 ( $n_{2}$ ), crossing the planar interface $y = 0$ at point $P = (x, 0)$ . Find the value of $x$ that makes the optical path length stationary.

Hint

Write the optical path length as $OPL = n_{1} x^{2} + a^{2} + n_{2} (d - x)^{2} + b^{2}$ . Differentiate with respect to $x$ and set the result to zero. The angles $θ_{1}$ and $θ_{2}$ satisfy $sin θ_{1} = x / x^{2} + a^{2}$ and $sin θ_{2} = (d - x) / (d - x)^{2} + b^{2}$ .

Answer

The optical path length is:

OPL (x) = n_{1} x^{2} + a^{2} + n_{2} (d - x)^{2} + b^{2}

Differentiate and set to zero:

\frac{d OPL}{d x} = \frac{n _{1} x}{x ^{2} + a ^{2}} - \frac{n _{2} ( d - x )}{( d - x ) ^{2} + b ^{2}} = 0

From the geometry, $sin θ_{1} = x / x^{2} + a^{2}$ (opposite over hypotenuse in the upper triangle) and $sin θ_{2} = (d - x) / (d - x)^{2} + b^{2}$ (in the lower triangle). Therefore:

n_{1} sin θ_{1} = n_{2} sin θ_{2}

This is Snell's law. Fermat's principle thus provides a variational derivation: the actual ray path is the one for which the optical path length is stationary (in this case a minimum). The same approach can be extended to curved interfaces and continuously varying media.

Exercise 4 (medium, symbolic).

A ray propagates in a medium where the refractive index varies only in the $x$ -direction: $n = n (x)$ . Show that the ray equation $\frac{d}{d s} (n \frac{d r}{d s}) = \nabla n$ implies the Bouguer's formula $n (x) cos α = const$ , where $α$ is the angle between the ray and the $x$ -axis.

Hint

Write out the $y$ -component of the ray equation. Since $n$ depends only on $x$ , $\partial n / \partial y = 0$ . This gives a conserved quantity along the ray.

Answer

The ray equation has components:

\frac{d}{d s} (n \frac{d x}{d s}) = \frac{\partial n}{\partial x}, \frac{d}{d s} (n \frac{d y}{d s}) = \frac{\partial n}{\partial y} = 0

The $y$ -component integrates immediately:

n \frac{d y}{d s} = C = const

Now $\frac{d y}{d s} = sin α$ where $α$ is the angle between the ray and the $x$ -axis (since $d s$ is along the ray and $d y$ is the $y$ -component). Therefore:

n sin α = C

Alternatively, if $α$ is measured from the $y$ -axis (so $cos α = d y / d s$ ), then $n cos α = C$ . This is Bouguer's formula: the product $n sin α$ (or equivalently $n cos β$ where $β$ is measured from the other axis) is conserved along a ray in a planar-stratified medium.

Bouguer's formula is the continuous analogue of Snell's law for a stratified medium. In the limit of a single sharp interface between two uniform layers, it reduces exactly to $n_{1} sin θ_{1} = n_{2} sin θ_{2}$ .

Exercise 5 (medium, symbolic).

The transport equation $2\nabla A \cdot \nabla S + A \nabla^{2} S = 0$ can be rewritten as $\nabla \cdot (A^{2} \nabla S) = 0$ . Show this, and interpret the result as energy conservation.

Hint

Expand $\nabla \cdot (A^{2} \nabla S)$ using the product rule: $\nabla (A^{2}) \cdot \nabla S + A^{2} \nabla^{2} S$ . Then note that $\nabla (A^{2}) = 2 A \nabla A$ .

Answer

Compute the divergence:

\nabla \cdot (A^{2} \nabla S) = \nabla (A^{2}) \cdot \nabla S + A^{2} \nabla^{2} S = 2 A \nabla A \cdot \nabla S + A^{2} \nabla^{2} S

= A (2\nabla A \cdot \nabla S + A \nabla^{2} S) = A \times 0 = 0

where the parenthesised expression is zero by the transport equation. Therefore $\nabla \cdot (A^{2} \nabla S) = 0$ .

Physical interpretation. The intensity of the wave is proportional to $A^{2}$ (the squared amplitude), and the energy flux (Poynting vector direction for scalar waves) is along $\nabla S$ (the ray direction). So $J = A^{2} \nabla S$ is the energy flux density, and $\nabla \cdot J = 0$ says that energy is conserved: no energy is created or destroyed along a ray bundle. This is consistent with the interpretation of the transport equation as governing intensity conservation along ray tubes.

Exercise 6 (medium, symbolic).

A simple telescope (Keplerian design) consists of two converging lenses separated by the sum of their focal lengths: $d = f_{1} + f_{2}$ , where $f_{1}$ is the objective focal length and $f_{2}$ is the eyepiece focal length. Compute the ABCD matrix for this system and find the overall angular magnification.

Hint

The system is: free space $f_{1}$ + thin lens $f_{1}$ + thin lens $f_{2}$ + free space $f_{2}$ . Wait -- the lenses are at positions 0 and $d = f_{1} + f_{2}$ . So the sequence is: lens $f_{1}$ (objective) at the front, free space $d = f_{1} + f_{2}$ , lens $f_{2}$ (eyepiece) at the back. Multiply the matrices in reverse order.

Answer

The optical elements in traversal order are: objective lens ( $f_{1}$ ), free-space propagation ( $d = f_{1} + f_{2}$ ), eyepiece lens ( $f_{2}$ ). The overall ABCD matrix is:

M = (1 - 1/ f_{2} 01) (10 f_{1} + f_{2} 1) (1 - 1/ f_{1} 01)

First, multiply the middle and right matrices:

(10 f_{1} + f_{2} 1) (1 - 1/ f_{1} 01) = (1 - (f_{1} + f_{2}) / f_{1} - 1/ f_{1} f_{1} + f_{2} 1) = (- f_{2} / f_{1} - 1/ f_{1} f_{1} + f_{2} 1)

Now multiply by the left matrix:

(1 - 1/ f_{2} 01) (- f_{2} / f_{1} - 1/ f_{1} f_{1} + f_{2} 1) = (- f_{2} / f_{1} 1/ f_{1} - (- f_{2} / f_{1}) (- 1/ f_{2}) f_{1} + f_{2} - (f_{1} + f_{2}) / f_{2} + 1)

The $(2, 1)$ entry: $1/ f_{1} + 1/ f_{2} \cdot 1/ f_{1} - 1/ f_{1}$ ... Let me recompute:

$(2, 1) = (- 1/ f_{2}) (- f_{2} / f_{1}) + (1) (- 1/ f_{1}) = 1/ f_{1} - 1/ f_{1} = 0$

$(2, 2) = (- 1/ f_{2}) (f_{1} + f_{2}) + 1 = - f_{1} / f_{2} - 1 + 1 = - f_{1} / f_{2}$

So:

M = (- f_{2} / f_{1} 0 f_{1} + f_{2} - f_{1} / f_{2})

The output angle is $θ_{out} = (- f_{1} / f_{2}) θ_{in}$ , so the angular magnification is $M_{θ} = - f_{1} / f_{2}$ . The minus sign indicates the image is inverted. For a telescope with $f_{1} = 100$ cm and $f_{2} = 2$ cm, the angular magnification is $- 50$ : objects appear 50 times larger but upside down.

The $(1, 1)$ element $- f_{2} / f_{1}$ is the transverse (linear) magnification, which is small: the telescope compresses the image in the transverse direction while expanding it angularly.

Exercise 7 (hard, symbolic).

Derive the lens maker's equation for a thin lens made of material with refractive index $n$ immersed in air ( $n_{air} = 1$ ). The lens has two spherical surfaces with radii of curvature $R_{1}$ and $R_{2}$ (positive if the centre of curvature is to the right of the surface). Work in the paraxial approximation.

Hint

Apply Snell's law at each surface in the paraxial approximation ( $sin θ \approx θ$ ). A ray at height $y$ hitting a surface of radius $R$ makes an angle $y / R$ with the normal. The deflection at each surface is $(n - 1) y / R$ . Add the deflections from both surfaces.

Answer

Consider a ray parallel to the axis at height $y$ , entering the lens from the left.

First surface (radius $R_{1}$ ). The normal to the surface at height $y$ makes angle $α_{1} \approx y / R_{1}$ with the axis (paraxial). The ray hits the surface at angle of incidence $α_{1}$ . Applying Snell's law in the paraxial approximation: the refraction angle satisfies $n θ_{r} \approx 1 \cdot α_{1}$ , so the ray is deflected by $Δ θ_{1} = (n - 1) y / R_{1}$ .

Second surface (radius $R_{2}$ ). The ray now hits the second surface at height $y$ (thin lens: negligible change in height). The normal makes angle $α_{2} \approx y / R_{2}$ with the axis. At this surface the ray goes from $n$ to $1$ , so by Snell's law the ray is deflected by $Δ θ_{2} = (1 - n) y / R_{2} = - (n - 1) y / R_{2}$ .

Total deflection. The combined angular deflection is:

Δ θ = Δ θ_{1} + Δ θ_{2} = (n - 1) \frac{y}{R _{1}} - (n - 1) \frac{y}{R _{2}} = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}}) y

A thin lens of focal length $f$ deflects a ray at height $y$ by $Δ θ = - y / f$ (from the ABCD matrix: $θ_{out} = θ_{in} - y / f$ ; for a ray entering parallel $θ_{in} = 0$ , so $θ_{out} = - y / f$ ). Therefore:

\frac{1}{f} = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})

This is the lens maker's equation. Examples: a symmetric biconvex lens ( $R_{1} = R$ , $R_{2} = - R$ ) gives $1/ f = 2 (n - 1) / R$ . A plano-convex lens ( $R_{1} = R$ , $R_{2} = \infty$ ) gives $1/ f = (n - 1) / R$ .

The sign convention for $R$ matters: $R > 0$ if the centre of curvature is to the right (surface convex toward the incoming light), $R < 0$ if the centre is to the left (concave toward the incoming light).

Exercise 8 (hard, symbolic).

A graded-index (GRIN) fibre has a parabolic refractive index profile $n (ρ) = n_{0} (1 - α^{2} ρ^{2} /2)$ where $ρ = x^{2} + y^{2}$ is the radial distance from the axis and $α$ is a small positive constant. Show that meridional rays (rays in a plane containing the axis) satisfy the equation of a harmonic oscillator, and find the period of oscillation.

Hint

Use the paraxial ray equation $d / d z (n d ρ / d z) = \partial n / \partial ρ$ . Approximate $n \approx n_{0}$ (constant) on the left side (since $α$ is small) and compute $\partial n / \partial ρ$ on the right.

Answer

The paraxial ray equation for a meridional ray (say in the $x$ - $z$ plane) is:

\frac{d}{d z} (n \frac{d x}{d z}) = \frac{\partial n}{\partial x}

Since $α$ is small, $n \approx n_{0}$ on the left side (the variation in $n$ contributes only higher-order corrections):

n_{0} \frac{d ^{2} x}{d z ^{2}} = \frac{\partial n}{\partial x}

Compute $\partial n / \partial x$ from $n = n_{0} (1 - α^{2} (x^{2} + y^{2}) /2)$ . For a ray in the $x$ - $z$ plane, $y = 0$ :

\frac{\partial n}{\partial x} = - n_{0} α^{2} x

Therefore:

\frac{d ^{2} x}{d z ^{2}} = - α^{2} x

This is the harmonic oscillator equation with solution:

x (z) = x_{0} cos (α z) + \frac{x _{0}^{'}}{α} sin (α z)

where $x_{0} = x (0)$ and $x_{0}^{'} = d x / d z ∣_{z = 0}$ are the initial position and slope of the ray. The period (the distance after which the ray returns to its original position and slope) is:

Λ = \frac{2 π}{α}

Every meridional ray, regardless of initial conditions, oscillates with the same period $Λ$ . This means all rays launched at the same $z$ -position refocus at $z = Λ/2, Λ, 3Λ/2, \dots$ . This is why parabolic GRIN fibres are used in communications: the equal period ensures that pulses do not spread due to different path lengths (no intermodal dispersion).

For non-parabolic index profiles, different rays have different periods, causing modal dispersion that limits bandwidth.

Lean formalization Intermediate+

Mathlib has no formalisation of the eikonal equation, the ray equation, Fermat's principle, ABCD matrices, the lens maker's formula, or the Hamiltonian optico-mechanical analogy. The eikonal equation is a Hamilton-Jacobi PDE whose characteristics give the rays; this requires the method of characteristics for first-order nonlinear PDEs, which is not in Mathlib. lean_status: none.

References and further reading Intermediate+

Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2017), Ch. 9.2-9.3.
Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999), Ch. 3.3.
Hecht, E., Optics, 5th ed. (Pearson, 2017), Ch. 4-5 (geometrical optics), Ch. 6 (ABCD matrices).

Advanced results Master

Hamilton's optico-mechanical analogy. The eikonal equation $∣\nabla S ∣^{2} = n^{2}$ and the ray equation $\frac{d}{d s} (n \frac{d r}{d s}) = \nabla n$ have exact counterparts in classical mechanics 09.04.09. The Hamilton-Jacobi equation for a particle of mass $m$ and energy $E$ in potential $V (r)$ is:

∣\nabla W ∣^{2} = 2 m (E - V (r))

The mechanical eikonal $W$ corresponds to the optical eikonal $S$ ; the momentum $p = 2 m (E - V)$ corresponds to the refractive index $n$ ; and the particle trajectories (solutions of Hamilton's equations) correspond to the light rays (solutions of the ray equation). The correspondence table is:

Optics	Mechanics
Eikonal $S$	Hamilton's principal function $W$
Refractive index $n$	Momentum magnitude $p = 2 m (E - V)$
Ray equation	Hamilton's equations / Newton's second law
Fermat's principle $δ \int n d s = 0$	Maupertuis's principle $δ \int p d s = 0$
Wavelength $λ$	Planck's constant $ℏ$

Hamilton discovered this analogy in 1833-1834. It was later recognised that the eikonal expansion $u = A e^{i k_{0} S}$ is identical in structure to the WKB approximation $ψ = A e^{iW /ℏ}$ in quantum mechanics, and the semiclassical limit $ℏ \to 0$ of quantum mechanics corresponds to the short-wavelength limit $λ \to 0$ of wave optics. This analogy was the historical motivation for de Broglie's matter waves and Schrodinger's wave equation.

Spherical aberration. The paraxial approximation assumes all rays make small angles with the axis. For a lens with spherical surfaces, rays at large heights $y$ (marginal rays) are refracted more strongly than the paraxial formula predicts. A ray at height $y$ is focused at distance:

\frac{1}{f ( y )} \approx \frac{1}{f} + \frac{y ^{2}}{f ^{3}} \cdot \frac{n ^{2}}{( n - 1 ) ^{2}} [\frac{n + 2}{n ( n - 1 ) ^{2}} \cdot \frac{1}{R _{1}^{2}} - \dots]

The focal length depends on $y$ : marginal rays focus closer to the lens than paraxial rays. The transverse ray aberration (distance from the paraxial focal point) scales as $y^{3}$ for a thin spherical lens. This is third-order (Seidel) spherical aberration. It can be reduced by using aspheric surfaces or by choosing the lens shape that minimises aberration for a given object distance (the "best-form" lens).

Chromatic aberration. The refractive index $n$ depends on wavelength (dispersion). Blue light has higher $n$ than red light in most glasses, so a converging lens focuses blue light at a shorter distance than red light. The longitudinal chromatic aberration is the difference in focal length between two wavelengths:

Δ f = - f \cdot \frac{Δ n}{n - 1}

where $Δ n = n_{blue} - n_{red}$ . An achromatic doublet corrects this by combining a converging lens (crown glass, low dispersion) with a diverging lens (flint glass, high dispersion). The two elements have equal and opposite chromatic aberrations, cancelling to first order while maintaining a net positive power.

Graded-index (GRIN) media. In a medium with $n = n (r)$ , the ray equation $\frac{d}{d s} (n \frac{d r}{d s}) = \nabla n$ describes curved ray paths. For the parabolic profile $n (ρ) = n_{0} 1 - α^{2} ρ^{2}$ (which approximates the more realistic $n (ρ) = n_{0} (1 - α^{2} ρ^{2} /2)$ for small $ρ$ ), the ray equation gives sinusoidal trajectories with period $Λ = 2 π / α$ . GRIN lenses (flat pieces of graded-index glass) can focus light without curved surfaces, and GRIN fibres are the standard transmission medium in optical communications because the parabolic profile equalises the path lengths of different modes, minimising intermodal dispersion.

Connection to semiclassical mechanics. The eikonal expansion can be carried to higher order:

u = e^{i k_{0} S} (A_{0} + \frac{A _{1}}{i k _{0}} + \frac{A _{2}}{( i k _{0} ) ^{2}} + \dots)

This is identical to the WKB expansion in quantum mechanics. The leading order gives the eikonal (Hamilton-Jacobi) equation, the next order gives the transport equation (probability current conservation), and higher orders give corrections that account for diffraction, tunnelling, and caustics. The semiclassical limit $k_{0} \to \infty$ is the same mathematical limit as $ℏ \to 0$ in quantum mechanics. The Keller-Maslov index and the Gutzwiller trace formula, which count classical periodic orbits in the semiclassical quantisation of chaotic systems, have direct optical analogues in the phase shifts accumulated by rays at caustics.

Full proof set Master

Proposition (Fermat's principle implies the ray equation). The Euler-Lagrange equation for the functional $I [r] = \int_{A}^{B} n (r) d s$ with $d s = ∣ d r ∣$ is the ray equation.

Proof. Parametrise the curve by arc length $s$ so that $∣ d r / d s ∣ = 1$ . The Lagrangian is $L = n (r) \overset{x}{˙}^{2} + \overset{y}{˙}^{2} + \overset{z}{˙}^{2}$ where dots denote $d / d s$ . Since $∣ d r / d s ∣ = 1$ , the Lagrangian simplifies to $L = n (r)$ .

However, it is more convenient to use an arbitrary parameter $τ$ and write $I = \int n ∣ \dot{r} ∣ d τ$ where dots now denote $d / d τ$ . The Euler-Lagrange equation is:

\frac{d}{d τ} \frac{\partial}{\partial r ˙} (n ∣ \dot{r} ∣) = \frac{\partial}{\partial r} (n ∣ \dot{r} ∣)

Since $\frac{\partial}{\partial r ˙} ∣ \dot{r} ∣ = \dot{r} /∣ \dot{r} ∣$ :

\frac{d}{d τ} (n \frac{r ˙}{∣ r ˙ ∣}) = ∣ \dot{r} ∣ \nabla n

Choose $τ = s$ (arc length) so that $∣ \dot{r} ∣ = 1$ and $\dot{r} = d r / d s$ :

\frac{d}{d s} (n \frac{d r}{d s}) = \nabla n

This is the ray equation. $□$

Proposition (Fermat's principle gives Snell's law). For a planar interface between media $n_{1}$ and $n_{2}$ , the stationary optical path condition gives $n_{1} sin θ_{1} = n_{2} sin θ_{2}$ .

Proof. (See Exercise 3.) The optical path length is $OPL = n_{1} x^{2} + a^{2} + n_{2} (d - x)^{2} + b^{2}$ . Setting $d (OPL) / d x = 0$ and recognising the geometric sines gives $n_{1} sin θ_{1} = n_{2} sin θ_{2}$ . The second derivative is positive for $n_{1}, n_{2} > 0$ , confirming this is a minimum. $□$

Proposition (Determinant of ABCD matrices). For any ABCD matrix representing a sequence of optical elements in a medium with entrance index $n_{1}$ and exit index $n_{2}$ , the determinant satisfies $det M = n_{1} / n_{2}$ .

Proof. Each individual element satisfies this: free-space propagation ( $det = 1$ ), thin lens ( $det = 1$ ), and flat interface ( $det = n_{1} / n_{2}$ ). For a sequence of $N$ elements: $det (M_{N} \dots M_{1}) = \prod_{i} det (M_{i})$ . For the internal elements, each entrance index equals the exit index of the previous element, so the product telescopes to $n_{1}^{(first)} / n_{N}^{(last)}$ . In a uniform medium, all indices are equal and $det M = 1$ . $□$

Connections Master

Electromagnetic waves 10.04.02 provide the wave equation from which the eikonal equation is derived in the short-wavelength limit.
Diffraction 10.08.03 is the complementary regime where the wavelength is not small; the Kirchhoff integral reduces to geometrical optics in the short-wavelength limit, and the eikonal expansion provides the systematic connection.
Hamiltonian mechanics 09.04.09 shares the mathematical structure of ray optics through the optico-mechanical analogy: the eikonal equation is a Hamilton-Jacobi equation, and the ray equation is Hamilton's equations.
Scalar wave equation in acoustics and quantum mechanics obeys the same eikonal/WKB asymptotic expansion, so the results of this unit apply to semiclassical mechanics, seismic ray theory, and underwater acoustics.
Semiclassical mechanics uses the WKB method, which is the quantum-mechanical analogue of the eikonal expansion; the classical limit $ℏ \to 0$ corresponds to the ray-optics limit $λ \to 0$ .

Historical and philosophical context Master

The law of reflection was known in antiquity (Euclid, c. 300 BCE). Snell's law of refraction was discovered experimentally by Thomas Harriot (1601) and Willebrord Snellius (1621), and published by Descartes (1637) using a particle model of light.

Fermat's principle (1662) was the first variational principle in physics. Fermat postulated that light takes the path of least time (not least distance), from which he derived Snell's law. This was controversial: the Cartesian school believed light travels faster in denser media, while Fermat's principle requires light to travel slower (which is correct). Huygens (1678) confirmed Fermat's prediction using the wave theory.

Hamilton's optico-mechanical analogy (1833-1834) was developed before quantum mechanics existed. Hamilton showed that the equations of ray optics and classical mechanics have identical mathematical structure, with the eikonal playing the role of the action. This was a mathematical curiosity until de Broglie (1924) proposed that particles have wave-like properties with wavelength $λ = h / p$ , and Schrodinger (1926) developed wave mechanics by literally replacing the refractive index in the eikonal equation with the momentum and interpreting the result as a wave equation. The Hamiltonian analogy is the historical bridge between classical and quantum physics.

Kirchhoff (1882) showed that geometrical optics emerges from the wave equation as the short-wavelength limit, using the asymptotic expansion that Sommerfeld and Runge (1911) formalised as the eikonal equation. This established geometrical optics as an approximation to wave optics, valid when $λ$ is small compared to all relevant length scales.

Luneburg (1964) and Kline and Kay (1965) developed the modern asymptotic theory of wave propagation, extending the eikonal expansion to higher orders and treating caustics, evanescent waves, and Gaussian beams systematically.

Bibliography Master

Primary sources

Fermat, P. de, "Synthesis ad refractiones" (1662), in Oeuvres de Fermat, Vol. 1 (1891).
Hamilton, W. R., "On a general method of expressing the paths of light, and of the planets, by the coefficients of a characteristic function," Dublin Univ. Review 1, 795-826 (1833).
Sommerfeld, A. and Runge, J., "Anwendung der Vektorrechnung auf die Grundlagen der geometrischen Optik," Ann. Phys. 340, 277-298 (1911).
Luneburg, R. K., Mathematical Theory of Optics (Univ. of California Press, 1964).
Kline, M. and Kay, I. W., Electromagnetic Theory and Geometrical Optics (Wiley, 1965).

Textbooks and monographs

Born, M. and Wolf, E., Principles of Optics, 7th ed. (Cambridge, 1999), Ch. 3-4.
Hecht, E., Optics, 5th ed. (Pearson, 2017).
Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999), Ch. 3.3.
Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2017), Ch. 9.2-9.3.
Guenther, R. D., Modern Optics (Wiley, 1990).
Stavroudis, O. N., The Optics of Rays, Wavefronts, and Caustics (Academic Press, 1972).