Electromagnetic Waves and Optics

Contents

Plane electromagnetic waves
The speed of light
Reflection and refraction
Polarization
Interference
Diffraction
Geometrical optics
Dispersion

This chapter develops the wave solutions of the Maxwell system of The Maxwell Equations: plane waves, polarization, energy transport, and the two great limits of the theory—physical optics, where interference and diffraction rule, and geometrical optics, the short-wavelength limit whose mechanics-side twin was met in Hamilton–Jacobi Theory and the Optical–Mechanical Analogy. Optics is far older than electrodynamics, so the chapter also runs backwards through history: two centuries of measured optical phenomena—refraction, polarization, interference, diffraction, dispersion, and the speed of light itself—were already quantitative science before Maxwell explained what was waving. Each phenomenon is stated with its founding measurement; the wave-optics experiments are treated in full apparatus detail in Experiment: Wave Optics, and the failure of the aether reading of these waves in Experiments: Light, the Aether, and Time.

Derivation pending.

Electromagnetic Waves and Optics: all derivations of this chapter are pending.

Plane electromagnetic waves

Monochromatic plane waves

[Reserved: transversality, the mutual orthogonality of \(\vect{E}\), \(\vect{B}\) and the wavevector, the ratio \(E/B=c\), and the dispersion relation \(\omega=c\abs{\vect{k}}\) in vacuum [Maxwell:1865] [Jackson:1999]; the electromagnetic spectrum from radio to gamma, with Hertz's waves (The Maxwell Equations) as its first artificial band; superposition and wave packets, with the Fourier machinery of Fourier Analysis and Integral Transforms.]

Energy and momentum transport

[Reserved: time-averaged Poynting flux, intensity, and radiation pressure for plane waves, specialising the conservation laws of The Maxwell Equations [Poynting:1884]; irradiance in SI units and the link to photometric practice.]

The speed of light

Astronomical determinations

[Reserved: Rømer's inference of a finite speed from the eclipse timings of Io [Roemer:1676], and Bradley's stellar aberration, which gave the first terrestrial-independent ratio of \(c\) to the orbital speed of the Earth [Bradley:1728]; aberration reappears as a relativity test in Experiments: Light, the Aether, and Time.]

Phenomenon 61.1 (Light propagates at a finite speed).

The eclipses of Jupiter's innermost satellite do not recur at strictly equal intervals as observed from the Earth. They fall progressively late through the part of the year in which the Earth is receding from Jupiter, and progressively early through the part in which it is approaching, and the discrepancy accumulated across half a year amounts to many minutes — far more than any irregularity of the satellite's orbit could produce, and correlated with the Earth's position rather than with Jupiter's [Roemer:1676]. Independently, the apparent position of every star describes a small annual ellipse whose angular size is the same for all stars regardless of their distance [Bradley:1728], which a parallax cannot be. Light therefore takes time to travel, and the second observation fixes the ratio of the Earth's orbital speed to the speed of light.

Derivation. Let the satellite emerge from its planet's shadow at strictly periodic instants \(t_{k}\), separated by the true orbital period \(T\), and let \(d(t)\) be the distance from the event to the observer. The emergence is seen at \(t_{k}+d(t_{k})/c\), so the observed interval between successive emergences is

\begin{equation}\tag{61.1} T_{\text{obs}}=T+\frac{d(t_{k+1})-d(t_{k})}{c}\ec \end{equation}

which exceeds \(T\) while the observer recedes and falls short of it while the observer approaches. Summing Equation (61.1) over all the eclipses of one recession, the intermediate distances cancel in pairs and the total accumulated lag is

\begin{equation}\tag{61.2} \Delta t_{\text{total}}=\frac{\Delta d}{c}\ec \end{equation}

\(\Delta d\) being the total increase in the Earth–Jupiter distance, which is close to the diameter of the Earth's orbit because Jupiter moves slowly by comparison. A finite accumulated lag therefore means a finite \(c\), and Equation (61.2) determines it from two measurable lengths and one measurable time. An infinite speed predicts strictly equal intervals, which is what is refuted.

Aberration is the same finiteness read as an angle: a telescope carried transversely at speed \(v\) must be tilted forward by \(\tan\theta=v/c\) for the light entering the objective to reach the eyepiece, so the apparent displacement is common to all stars and annual in period — and it measures \(v/c\), not \(c\) alone.

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Terrestrial measurements

[Reserved: Fizeau's toothed-wheel measurement over the Suresnes–Montmartre baseline [Fizeau:1849]; Foucault's rotating mirror, which both measured \(c\) in air and showed light travels slower in water, deciding against the corpuscular theory [Foucault:1850], and his 1862 absolute determination [Foucault:1862]; Michelson's 1879 Annapolis measurement [Michelson:1880] and his Mount Wilson–San Antonio triangulation [Michelson:1927].]

Phenomenon 61.2 (Light travels more slowly in a denser medium).

Light travels more slowly in water than in air. The rotating-mirror method makes the comparison directly, sending the two beams over equal paths and reading the difference as a displacement of the returned image, and the speed in water is lower by very nearly the factor \(n\) by which water refracts [Foucault:1850]. The measurement is decisive rather than merely quantitative: the wave account requires the speed to fall in the more refracting medium, and the emission account requires it to rise.

Derivation. In a transparent non-magnetic medium the macroscopic Maxwell equations have the same form as in vacuum with \(\epsilon_{0}\) replaced by \(\epsilon=\epsilon_{r}\epsilon_{0}\) (Electrodynamics in Matter), so the wave equation derived in The Maxwell Equations carries a propagation speed

\begin{equation}\tag{61.3} v=\frac{1}{\sqrt{\mu_{0}\epsilon}}=\frac{c}{\sqrt{\epsilon_{r}}} =\frac{c}{n}\ec\qquad n:=\sqrt{\epsilon_{r}}\ge1\ep \end{equation}

That the same \(n\) is the refractive index of Snell's law follows from Phenomenon 61.3: matching the wavefronts across the interface gives \(\sin\theta_{1}/\sin\theta_{2}=v_{1}/v_{2}\), so a medium in which the ray bends towards the normal is one in which the wave is slower.

The emission theory reverses the conclusion from the same data. There the ray bends because the interface pulls the corpuscle normally, leaving the tangential velocity component unchanged; then \(v_{1}\sin\theta_{1}=v_{2}\sin\theta_{2}\), so \(v_{2}/v_{1}=\sin\theta_{1}/\sin\theta_{2}=n>1\) and light must go faster in water. The two predictions differ by the factor \(n^{2}\approx1.8\) for water and are not reconcilable; the measurement gives the wave value.

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From measured constant to defined constant

[Reserved: the laser frequency-times-wavelength determination of Evenson and collaborators [Evenson:1972], the 1983 fixing of \(c\) at exactly \(299792458\,\mathrm{m}/\mathrm{s}\), and the consequent redefinition of the metre [BIPM:2019]; what it means, in the framework of Measurement, SI Units, and the Theory of Errors, for a constant to become a definition.]

Reflection and refraction

Snell's law and Fermat's principle

[Reserved: the sine law of refraction, found by Snel around 1621 and first printed in Descartes' Dioptrique [Descartes:1637]; Fermat's derivation of the same law from least time, in the 1662 letters to Cureau de la Chambre [Fermat:1894]; Huygens' wavefront construction [Huygens:1690]; total internal reflection and the evanescent wave [Born:1999].]

Phenomenon 61.3 (The sine law of refraction).

A ray crossing the plane boundary between two transparent media is bent, and the ratio of the sines of the angles of incidence and refraction, both measured from the normal, is the same for every angle of incidence and depends only on the pair of media [Descartes:1637]:

\begin{equation}\tag{61.4} n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2}\ep \end{equation}

The constancy of the ratio is the content of the observation — the angles themselves are not proportional, and the older tables that assumed they were fail at large incidence.

Derivation. Fermat's principle: of all paths from a point \(A\) in medium \(1\) to a point \(B\) in medium \(2\), light takes the one of stationary transit time [Fermat:1894]. Place the interface along the plane \(y=0\), let \(A=(0,a)\) and \(B=(d,-b)\) with \(a,b>0\), and let the path cross at \((x,0)\). The optical time is

\begin{equation}\tag{61.5} t(x)=\frac{n_{1}}{c}\sqrt{x^{2}+a^{2}} +\frac{n_{2}}{c}\sqrt{(d-x)^{2}+b^{2}}\ec \end{equation}

using \(v_{i}=c/n_{i}\) from Equation (61.3). Setting \(\dd t/\dd x=0\),

\begin{equation}\tag{61.6} \frac{n_{1}x}{\sqrt{x^{2}+a^{2}}} =\frac{n_{2}(d-x)}{\sqrt{(d-x)^{2}+b^{2}}}\ec \end{equation}

and the two quotients are precisely \(\sin\theta_{1}\) and \(\sin\theta_{2}\), the angles being measured from the normal \(y\)-axis. This is Equation (61.4). The second derivative of Equation (61.5) is positive, so the stationary point is a minimum, and it is unique. Huygens' construction gives the same law geometrically, by requiring the wavefronts on the two sides to keep step along the interface [Huygens:1690], and this is the version that identifies \(n\) with \(c/v\) rather than leaving it an empirical constant of the pair of media.

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The Fresnel equations

[Reserved: amplitude reflection and transmission coefficients for the two polarizations, derived by Fresnel from his transverse-wave theory in the memoir read in 1823 [Fresnel:1832]; Brewster's angle of complete polarization by reflection, established empirically by Brewster [Brewster:1815]; phase shifts on reflection and their interferometric consequences; the electromagnetic rederivation from the boundary conditions of The Maxwell Equations [Jackson:1999].]

Phenomenon 61.4 (Complete polarization at the Brewster angle).

For every transparent dielectric there is one angle of incidence, and one only, at which the reflected light is completely linearly polarized, its electric vector lying in the surface. The tangent of that angle is the refractive index of the material [Brewster:1815]:

\begin{equation}\tag{61.7} \tan\theta_{B}=n\ep \end{equation}

Equivalently, at \(\theta_{B}\) the reflected and refracted rays are at right angles, as follows at once from Equation (61.4). The reflectance for light polarized in the plane of incidence falls smoothly to exactly zero there and rises again beyond it, so the vanishing is a genuine zero and not a minimum.

Derivation pending.

Brewster's angle: derive the Fresnel amplitude coefficients for the two polarizations from the interface conditions on the tangential fields, and show that the coefficient for light polarized in the plane of incidence has a simple zero exactly where the reflected and refracted directions are perpendicular, which by the sine law is at the angle whose tangent is the refractive index.

Polarization

Discovery and phenomenology

[Reserved: double refraction of Iceland spar, reported by Bartholin [Bartholin:1669] and given a wave construction by Huygens [Huygens:1690]; Malus's discovery of polarization by reflection and his cosine-squared intensity law [Malus:1809]; Brewster's law relating the polarizing angle to the refractive index [Brewster:1815].]

Phenomenon 61.5 (Malus's law).

Light that has been reflected from glass at a suitable angle, or transmitted through a crystal of Iceland spar, acquires a property that depends on the orientation of the beam about its own axis: passed through a second such device, it is transmitted in full at one orientation, extinguished at the orientation at right angles to it, and in between the transmitted intensity follows

\begin{equation}\tag{61.8} I(\theta)=I_{0}\cos^{2}\theta \end{equation}

with \(\theta\) the angle between the two devices [Malus:1809]. Ordinary light shows no such dependence, which is why the property had to be discovered rather than assumed.

Derivation. Take the beam leaving the first polarizer to be linearly polarized with electric field amplitude \(\vect{E}_{0}=E_{0}\hat{\vect{e}}_{1}\). An ideal analyser transmits the component of the field along its own axis \(\hat{\vect{e}}_{2}\) and rejects the orthogonal component, so the transmitted amplitude is

\begin{equation}\tag{61.9} E=\vect{E}_{0}\cdot\hat{\vect{e}}_{2}=E_{0}\cos\theta\ec \end{equation}

\(\theta\) being the angle between the two axes. Intensity is proportional to the time-averaged square of the field — it is \(\epsilon_{0}cE^{2}/2\) for a plane wave in vacuum, by The Maxwell Equations — so squaring Equation (61.9) gives Equation (61.8). The whole content of the derivation is that the observable is the square of a projected vector: that is why the law is \(\cos^{2}\theta\) rather than \(\abs{\cos\theta}\), and it is the first quantitative evidence that the optical disturbance is a directed quantity transverse to the ray.

∎

Mathematical description

[Reserved: linear, circular and elliptical polarization of a transverse plane wave; the Stokes parameters and the description of partially polarized light [Stokes:1852]; Jones vectors for fully polarized beams [Born:1999]; the measurement of the angular momentum carried by circularly polarized light [Beth:1936].]

Birefringence and wave plates

[Reserved: ordinary and extraordinary rays, the index ellipsoid, quarter- and half-wave plates, and polarimetry [Born:1999]; induced birefringence is deferred to the electro- and magneto-optics of Electrodynamics in Matter.]

Interference

Young's double slit

[Reserved: Young's demonstration that light added to light can give darkness, and his measurement of wavelengths from fringe spacings [Young:1804]—the decisive evidence for the wave theory; the sinusoidal two-source pattern and its visibility; apparatus detail in Experiment: Wave Optics, and the single-quantum version of the experiment in Matter Waves.]

Phenomenon 61.6 (Two-beam interference).

Light from a single small source, divided into two beams by a pair of narrow apertures a distance \(d\) apart and allowed to fall on a screen at distance \(D\gg d\), produces not two patches of light but a set of equally spaced bright and dark bands. Blocking either aperture destroys the bands and leaves a smooth illumination, so the dark bands are places where light added to light gives darkness. The band spacing is

\begin{equation}\tag{61.10} \Delta y=\frac{\lambda D}{d}\ec \end{equation}

proportional to \(D\) and inversely proportional to \(d\), and this is what first made the wavelength of light a measured quantity [Young:1804].

Derivation. Let the two apertures lie at \(\pm d/2\) on a line perpendicular to the axis, and let \(y\) measure position on the screen from the axis. The path lengths from the two apertures to the point \(y\) differ, for \(d\ll D\) and \(\abs{y}\ll D\), by

\begin{equation}\tag{61.11} \delta=\sqrt{D^{2}+\left(y+\tfrac{d}{2}\right)^{2}} -\sqrt{D^{2}+\left(y-\tfrac{d}{2}\right)^{2}} \simeq\frac{yd}{D}\ec \end{equation}

by expanding each root to first order in \((y\pm d/2)^{2}/D^{2}\). Superposing two disturbances of equal amplitude and of the same frequency, whose phases differ by \(2\pi\delta/\lambda\), gives an intensity

\begin{equation}\tag{61.12} I(y)=4I_{1}\cos^{2}\!\left(\frac{\pi\delta}{\lambda}\right) =4I_{1}\cos^{2}\!\left(\frac{\pi d\,y}{\lambda D}\right)\ec \end{equation}

because the amplitudes, not the intensities, are what add. This is maximal where \(\delta\) is an integer multiple of \(\lambda\) and zero where it is a half-odd multiple, so consecutive maxima are separated by \(\Delta y=\lambda D/d\), which is Equation (61.10). Blocking one aperture removes the interference term and leaves the constant \(I_{1}\), as observed; and Equation (61.10) inverted, \(\lambda=d\,\Delta y/D\), is the measurement, since \(d\), \(\Delta y\) and \(D\) are all lengths that can be read off with a scale.

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Thin films and interferometers

[Reserved: thin-film and wedge interference, Newton's rings (described in the Opticks [Newton:1704]); division of amplitude and the Michelson interferometer, built for the aether-drift measurement [Michelson:1887] and treated as a null relativity test in Experiments: Light, the Aether, and Time; multiple-beam interference and the Fabry–Pérot étalon [Born:1999].]

Coherence

[Reserved: temporal and spatial coherence as the condition for visible fringes; the van Cittert–Zernike theorem relating source size to transverse coherence [vanCittert:1934] [Zernike:1938]; Michelson's stellar interferometer and the measured diameter of Betelgeuse [Michelson:1921]; quantum coherence of light is taken up in Quantum Optics and the Photon.]

Diffraction

Fresnel diffraction and the Arago spot

[Reserved: Fresnel's prize memoir on diffraction, combining Huygens' secondary wavelets with interference [Fresnel:1826]; Poisson's reductio—a bright spot at the centre of a circular shadow —observed by Arago and reported with the memoir [Fresnel:1826], converting the objection into a confirmation; Fresnel zones and the zone plate.]

Phenomenon 61.7 (The bright spot in the centre of a shadow).

A small opaque disc of smooth circular edge, illuminated by a point source, casts a shadow with a bright point at its exact centre; for an ideal disc the intensity there equals the intensity that would be found with the disc removed. The straight edge and the disc are also surrounded by systems of fringes whose positions can be measured on a micrometer and compared, one by one, with the values computed from the diffraction integral [Fresnel:1826]. Neither the bright centre nor the fringe positions has any account in a theory of rays.

Derivation pending.

The axial bright point: derive it from the Huygens–Fresnel superposition of secondary wavelets over the plane of the obstacle, showing that every point of the disc's rim is equidistant from the axial field point, so that the wavelets from the rim arrive exactly in phase whatever the radius of the disc, and evaluate the resulting on-axis amplitude for an ideal circular obstacle.

Fraunhofer diffraction and gratings

[Reserved: far-field diffraction as the Fourier transform of the aperture, with the machinery of Fourier Analysis and Integral Transforms; Fraunhofer's diffraction gratings and his wavelength standards [Fraunhofer:1821]; the single slit, the circular aperture, and the Rayleigh resolution criterion [Rayleigh:1879].]

Kirchhoff's scalar theory

[Reserved: the scalar diffraction integral derived from the wave equation with Kirchhoff's boundary values [Kirchhoff:1883], its internal inconsistency and the Rayleigh–Sommerfeld repair [Born:1999]; the validity conditions under which the earlier Fresnel and Fraunhofer treatments emerge as limits.]

Geometrical optics

The eikonal limit

[Reserved: the short-wavelength expansion of the wave equation, the eikonal equation \((\vect{\nabla}\mathcal{S})^{2}=n^{2}\), and the ray equation, named after Bruns's Das Eikonal [Bruns:1895]; Hamilton's characteristic-function optics [Hamilton:1828] and its mechanical twin, already developed in Hamilton–Jacobi Theory and the Optical–Mechanical Analogy; where the limit fails—edges and caustics [Born:1999].]

Images and instruments

[Reserved: mirrors, thin lenses and the paraxial imaging equations; the telescope and microscope as diffraction-limited instruments, tying the resolution limit of Section 61.6.2 to practice [Born:1999]; aberrations named and classified.]

Dispersion

The prism and the solar spectrum

[Reserved: Newton's experimentum crucis separating white light into unequally refrangible rays [Newton:1672] and its systematic account in the Opticks [Newton:1704]; Fraunhofer's dark lines in the solar spectrum and his measurement of their wavelengths [Fraunhofer:1817], the founding data of the spectroscopy taken up in Atomic Models and Spectra.]

Phenomenon 61.8 (Dispersion and the fixed lines of the solar spectrum).

A prism spreads white light into a band of colours in an invariable order, and a colour once separated is not decomposed further by a second prism, nor changed by transmission through matter: the refractive index is a function of the colour, and each colour is a simple constituent of the light rather than a modification produced by the glass [Newton:1672] [Newton:1704]. Examined at sufficient resolution, the spectrum of the Sun is crossed by a great number of fixed dark lines, always in the same places, whose refractive positions can be measured far more precisely than any colour can be named — which is what turns dispersion from a qualitative effect into an instrument [Fraunhofer:1817].

Derivation pending.

Dispersion: derive the frequency dependence of the refractive index from the response of bound charges to the field of the wave, obtaining normal dispersion away from resonance and anomalous dispersion with absorption near it, and recover the empirical Sellmeier form as its low-loss limit. The model belongs to the treatment of electrodynamics in matter, and the discreteness of the solar lines themselves is not classical at all and awaits the atomic chapters.

The rainbow

[Reserved: Descartes' ray-tracing of the primary and secondary bows [Descartes:1637]; Newton's account of their colours [Newton:1704]; supernumerary bows as an interference effect and Airy's caustic integral [Airy:1838]—the rainbow as the first quantitative test case where ray optics fails and wave optics is required.]

Phase velocity, group velocity, and media

[Reserved: normal and anomalous dispersion; the Sellmeier relation [Sellmeier:1871]; Rayleigh's distinction of group from phase velocity [Rayleigh:1877]; light in moving media—Fresnel's partial-drag coefficient [Fresnel:1818] and Fizeau's moving-water interferometry confirming it [Fizeau:1851], reread relativistically in Experiments: Light, the Aether, and Time; the microscopic theory of dispersion belongs to Electrodynamics in Matter.]