David Brewster

How to use this course

David Brewster is remembered by students for a single angle and by the public for a toy. Both memories are accurate and both are too small. Between 1811 and 1819 he carried out the most extensive experimental survey of polarized light of his era. He found the law of polarization by reflection, showed that strain makes glass doubly refracting, and sorted hundreds of crystals into optical classes that matched their geometry. He also edited an encyclopaedia, helped found the British Association for the Advancement of Science, wrote the standard Victorian life of Isaac Newton, and led two Scottish universities.

Brewster worked by measurement and generalisation rather than by theory, and he never accepted the wave theory of light that explained his own laws. Part II therefore pairs each of his empirical results with the modern derivation, usually due to Augustin-Jean Fresnel or later physicists. Each technical lesson opens with a paragraph for the general reader, then gives statements, derivations and exercises with folded solutions.

Timeline

Jedburgh and Edinburgh, 1781–1800

David Brewster was born on 11 December 1781 in Jedburgh, a small Border town in Roxburghshire. His father, James Brewster, was rector of the town’s grammar school and a respected teacher. His mother, Margaret Key, died in 1790, when he was eight. He was the third of six children, and three of his brothers, James, George and Patrick, became ministers of the Church of Scotland. The family expected David to follow them.

As a boy he built a telescope with the help of a local self-taught astronomer and instrument maker, James Veitch of Inchbonny. At twelve he matriculated at the University of Edinburgh, a normal age for a Scottish student then. He took his MA in 1800. In 1799 his fellow student Henry Brougham, later Lord Chancellor, drew him into the study of the diffraction of light, the subject that would occupy him for the rest of his life.

Preacher, tutor, editor, 1800–1811

After his MA Brewster studied divinity and was licensed to preach in the Church of Scotland. He found preaching an ordeal and did not seek a parish. Instead he supported himself as a private tutor and, increasingly, as a writer and editor. He wrote for the Edinburgh Magazine from 1799 and edited it in 1802 and 1803. In 1804 he published a history of freemasonry.

In 1807 he became editor of the Edinburgh Encyclopaedia, a vast project that appeared in parts from 1808 to 1830 and to which he contributed many of the scientific articles himself. The same year Marischal College, Aberdeen, gave him an honorary LLD. He was elected to the Royal Society of Edinburgh in 1808, and on 31 July 1810 he married Juliet Macpherson. They had several children; their son Charles drowned in 1828.

Editing gave Brewster a living, a wide reading of current science, and a lifelong habit of controversy in print. It did not give him a laboratory. His most important experiments were made at home, with crystals, glass plates, a lamp and the sky.

The polarization years, 1811–1819

In 1808 Étienne-Louis Malus in Paris discovered that light reflected from glass is polarized: viewed through a calcite crystal, its two images change in brightness as the crystal turns. Malus died in 1812. Brewster, then in his thirties and working largely alone in Scotland, took up the subject with great energy, measuring reflection from dozens of substances and the behaviour of light in crystals.

In a paper read to the Royal Society on 16 March 1815 he announced that the angle of complete polarization depends on the refractive index alone: its tangent equals the index (Lesson 04). The same year he was elected a Fellow of the Royal Society and received its Copley Medal. In 1815 and 1816 he showed that glass under pressure or unequal heating becomes doubly refracting and shows coloured fringes in polarized light (Lesson 07). From 1813 to 1818 he examined hundreds of crystals and found that they fall into two optical classes, with one or two axes of no double refraction, matching their crystallographic symmetry (Lesson 06). The Royal Society awarded him the Rumford Medal in 1818.

His work on multiple reflections between mirrors, begun in 1814, gave him the kaleidoscope in 1815 (Lesson 08). He patented it in 1817, but the design was so easy to copy that he made little money from it. In 1819 he founded the Edinburgh Philosophical Journal with the mineralogist Robert Jameson and became General Secretary of the Royal Society of Edinburgh.

Brewster’s law

For the general reader. Light reflected from water or glass is partly polarized, which is why polarizing sunglasses cut glare. At one particular angle the reflected light is completely polarized. Brewster found by measurement that this angle depends only on how strongly the material bends light: its tangent is the refractive index. Fresnel’s wave theory explained the law a few years later.

Brewster stated the law in the form “the index of refraction is the tangent of the angle of polarisation”. He found it by measuring polarizing angles for many substances, and observed that for each the reflected and refracted rays met at a right angle.

Derivation from the Fresnel equations

For non-magnetic media, the amplitude reflection coefficients for the field perpendicular (s) and parallel (p) to the plane of incidence are

where \(n_1\sin\theta_i=n_2\sin\theta_t\). (Sign conventions for \(r_p\) differ between textbooks; the reflectances \(R=|r|^2\) do not.) The coefficient \(r_p\) vanishes when \(\tan(\theta_i+\theta_t)\) is infinite, that is when \(\theta_i+\theta_t=\pi/2\). Then \(\sin\theta_t=\cos\theta_i\), and Snell’s law gives \(n_1\sin\theta_i=n_2\cos\theta_i\), which is Brewster’s law.

Physical picture

The reflected wave is radiated by dipoles in the second medium, oscillating along the field of the refracted wave. For p-polarization that field lies in the plane of incidence, perpendicular to the refracted ray. If the reflected ray is perpendicular to the refracted ray, it points along the dipole axis, and a dipole does not radiate along its own axis. No p-light is reflected.

Piles of plates and Brewster windows

For the general reader. One reflection at Brewster’s angle gives perfectly polarized but faint light. Brewster studied the light transmitted through a stack of glass plates, which loses a little of one polarization at each surface and none of the other. With enough plates the transmitted beam becomes strongly polarized. The same principle now keeps laser beams polarized.

Each plate has two surfaces, each passing p-light completely (Exercise 4.3) and s-light with transmittance \(1-R_s\). The model neglects light reflected back and forth between surfaces, which adds some s-light to the transmitted beam and lowers \(P\) somewhat. Brewster studied the polarization of light transmitted obliquely through such piles in his early papers.

Brewster windows

In a gas laser the gain tube is often closed by windows tilted at Brewster’s angle. Light polarized in the plane of incidence passes them with no reflection loss, while the other polarization loses about 15% per surface, far more than the gain can replace. The laser therefore oscillates only in p-polarization.

Biaxial crystals

For the general reader. A calcite crystal splits a beam of light in two, except along one special direction, its optic axis. Brewster discovered that many crystals have two such directions, not one, and that whether a crystal has none, one or two depends on the symmetry of its shape. This connected optics to crystallography and gave mineralogists a new identification tool.

The index ellipsoid

In a transparent non-magnetic crystal, \(\mathbf D=\varepsilon_0\boldsymbol\varepsilon\mathbf E\) with \(\boldsymbol\varepsilon\) a real symmetric tensor. In its principal axes \(\boldsymbol\varepsilon=\mathrm{diag}(n_\alpha^2,n_\beta^2,n_\gamma^2)\) with \(n_\alpha\le n_\beta\le n_\gamma\). The index ellipsoid is

For a wave travelling in direction \(\mathbf s\), cut the ellipsoid by the plane through the origin perpendicular to \(\mathbf s\). The section is an ellipse whose semi-axes are the two refractive indices for that direction, and whose axes give the two allowed directions of \(\mathbf D\). Double refraction disappears when the section is a circle.

Proof. Every central plane meets the \(xz\)-plane, and the \(y\)-axis, in lines whose radii bracket \(n_\beta\); a short argument shows that a circular section must have radius \(n_\beta\) and must contain the \(y\)-axis, whose radius is \(n_\beta\). A plane containing the \(y\)-axis and the unit vector \((\cos V,0,-\sin V)\) of the \(xz\)-plane is circular when that vector also has length \(n_\beta\) on the ellipsoid:

The normal to that plane is \((\sin V,0,\cos V)\), at angle \(V\) from \(z\). Solving for \(\tan^2V\) gives the formula; the sign choice \(\pm V\) gives two axes. ∎

Brewster’s classification

Symmetry forces the shape of \(\boldsymbol\varepsilon\). Cubic crystals (rock salt, diamond) are isotropic. Tetragonal, hexagonal and trigonal crystals (zircon, quartz, calcite) have one distinguished axis and are uniaxial. Orthorhombic, monoclinic and triclinic crystals (topaz, mica, aragonite) are biaxial. Brewster established this correspondence experimentally for hundreds of minerals between 1813 and 1819, before the tensor description existed. Fresnel’s wave-surface theory of 1821 gave the mathematical account.

Stressed glass and photoelasticity

For the general reader. Ordinary glass does not split light. Brewster found that squeezed, bent or unevenly heated glass does, and that viewed between crossed polarizers it shows bands of colour that map the strain. A century later engineers built transparent plastic models of bridges and gears and photographed these bands to find where they would break.

Stress-optic coefficients are customarily quoted in brewsters: \(1\ \text{brewster}=10^{-12}\ \mathrm{Pa}^{-1}\). Typical glasses have a few brewsters.

The polariscope

Place the plate between a polarizer and an analyzer crossed at \(90^\circ\). Let the principal stress direction make angle \(\varphi\) with the polarizer. Resolving the field along the two principal directions and recombining at the analyzer gives

The intensity vanishes where \(\sin2\varphi=0\), the isoclinics, where a principal direction is parallel to the polarizer; and where \(\delta=2\pi N\), the isochromatics, along which \(\sigma_1-\sigma_2\) is constant. In white light each wavelength goes dark at a different stress, producing the colours Brewster saw. Inserting quarter-wave plates (a circular polariscope) removes the \(\varphi\)-dependence, leaving \(I=I_0\sin^2(\delta/2)\).

The kaleidoscope

For the general reader. Two mirrors set at an angle produce many images of an object between them. Brewster worked out when these images join into a perfect, seamless symmetric pattern, and named his instrument from Greek words for beautiful form and viewing. Mathematically, a kaleidoscope is a picture of a reflection group.

Proof. In complex notation, reflection in the line at angle \(a\) is \(z\mapsto e^{2ia}\bar z\). Composing reflections at \(0\) and \(\theta\): \(z\mapsto\bar z\mapsto e^{2i\theta}z\), a rotation by \(2\theta\). If \(\theta=\pi/n\) the rotation has order \(n\), and with one reflection generates a group of \(2n\) elements; the \(2n\) images of the wedge of angle \(\pi/n\) fill \(2\pi\). ∎

Why must the angle divide \(180^\circ\) and not merely \(360^\circ\)? If \(\theta=2\pi/m\) with \(m\) odd, the rotation \(\rho_2\rho_1\) by \(4\pi/m\) has order \(m\), so the group has \(2m\) elements, but \(2m\) wedges of angle \(2\pi/m\) cover the turn twice. The images overlap inconsistently and an asymmetric object shows a seam. Brewster’s patent named \(18^\circ,20^\circ\) and \(22\tfrac12^\circ\), that is \(\pi/10,\pi/9,\pi/8\).

The stereoscope and binocular depth

For the general reader. Each eye sees the world from a slightly different place, and the brain turns the difference into depth. Charles Wheatstone demonstrated this in 1838 with a mirror stereoscope. In 1849 Brewster replaced the mirrors with lenses, producing a small hand-held viewer that, with the new photography, became one of the most popular optical devices of the century.

The convergence angle for distance \(Z\) is \(\gamma=2\arctan(b/2Z)\approx b/Z\). Differentiating, \(|d\gamma/dZ|\approx b/Z^2\), which is the formula. Depth resolution falls as \(1/Z^2\), so stereo vision matters most within a few tens of metres.

Brewster’s design

A stereo card carries two photographs taken from viewpoints a few centimetres apart. Brewster’s viewer places a convex half-lens before each eye. Each half-lens magnifies its picture and, because it is the off-centre part of a lens, also deflects light like a weak prism, so the eyes can fuse the two magnified images without strain. No instrument maker in Britain would take up the design, so he had it made in Paris by Jules Duboscq. Shown at the Great Exhibition of 1851, it caught the attention of Queen Victoria, and stereoscopes were soon sold by the hundred thousand. Brewster then disputed Wheatstone’s priority bitterly, crediting the principle to an Edinburgh teacher named Elliot.

Zoned lenses and absorption lines

For the general reader. A lens big enough to send a lighthouse beam far out to sea would be impossibly thick and heavy if made in one piece. Brewster proposed in 1811 to build it from separate rings of glass, each bending light like a prism. Fresnel independently designed and built such lenses in France. Brewster also studied how gases and crystals absorb particular colours, an early step toward spectroscopy.

Built-up lenses

A lens collimates light from a source at its focus. A ray leaving the focus at height \(h\) on a lens of focal length \(f\) must be turned through an angle \(\arctan(h/f)\). The middle of a plano-convex lens does this with gently curved glass; the outer parts need steep surfaces and great thickness. A zoned lens replaces each annulus by a thin ring whose surface has the right slope, keeping the deviation and discarding the bulk. Georges-Louis Leclerc de Buffon had proposed a stepped burning glass in one piece in 1748; Brewster in 1811 proposed building the zones separately. Fresnel’s lens, built from separate prisms, was first lit at the Cordouan lighthouse in 1823. Brewster pressed British lighthouse authorities to adopt such lenses from 1820 and later complained that Fresnel received the credit.

Absorption

In 1818–19 Brewster showed that some crystals absorb light differently according to its polarization and direction, the effect now called pleochroism. In 1833 and 1834 he found that nitrous acid gas (nitrogen dioxide) imprints many dark lines on a spectrum, and that some dark lines in the solar spectrum strengthen as the Sun sinks, so they are produced by the Earth’s atmosphere. In 1860, with the chemist John Hall Gladstone, he published a map of the solar spectrum separating these atmospheric lines from solar ones. Around the same time Gustav Kirchhoff and Robert Bunsen showed that the solar lines identify chemical elements.

Science, church and controversy, 1820–1859

Through the 1820s Brewster edited journals, finished the Encyclopaedia and wrote prolifically. He started the Edinburgh Journal of Science in 1824 and received the Keith Prize of the Royal Society of Edinburgh and, in 1830, the Royal Medal of the Royal Society. He shared with Charles Babbage a conviction that British science was neglected by the state and the universities, and in 1831 he was one of the chief promoters of the British Association for the Advancement of Science, whose first meeting was held at York that September. The same year he was knighted and received the Royal Guelphic Order. His Life of Sir Isaac Newton (1831) and Letters on Natural Magic (1832), addressed to his friend Sir Walter Scott, made him a well-known writer.

In optics he remained apart from the majority. He gave up the particle theory of light but never accepted the wave theory, and he argued in print with its defenders. He also fought priority disputes: with Fresnel’s supporters over lighthouse lenses, and with Wheatstone over the stereoscope. He was a friend and supporter of William Henry Fox Talbot, and in Scotland he encouraged the calotype photography of Robert Adamson and David Octavius Hill.

In 1838 he became Principal of the United College of St Salvator and St Leonard at St Andrews. His reforming energy and temper brought constant conflict with colleagues. In 1843, at the Disruption, he left the established Church of Scotland for the new Free Church, which made his position at St Andrews contentious. His wife Juliet died in 1850. In 1854 he published More Worlds than One, a reply to William Whewell defending the view that other planets are inhabited. His enlarged Memoirs of Newton (1855) drew on unpublished papers and remained the standard biography for a century.

Edinburgh, death, legacy, 1859–1868

In 1857, aged 75, Brewster married Jane Kirk Purnell in Nice. In 1859 he became Principal of the University of Edinburgh, where he had studied as a boy. In 1860 he published with John Hall Gladstone the map of the solar spectrum described in Lesson 10, and in 1864 he became President of the Royal Society of Edinburgh. He also wrote critically of Charles Darwin’s theory. He died of pneumonia on 10 February 1868 at Allerly, his house at Gattonside near Melrose, aged 86, and was buried in the grounds of Melrose Abbey.

Legacy

• Optics: Brewster’s angle and Brewster windows; the optical classification of crystals into isotropic, uniaxial and biaxial; pleochroism.
• Engineering: photoelasticity, used for stress analysis in models and for checking residual stress in glass; the unit of the stress-optic coefficient is named the brewster.
• Instruments: the kaleidoscope and the lenticular stereoscope; advocacy of built-up lenses for lighthouses.
• Institutions: the British Association; leadership of the universities of St Andrews and Edinburgh and of the Royal Society of Edinburgh.
• Honours: the Copley (1815), Rumford (1818) and Royal (1830) medals; foreign associate of the Institut de France (1849); the Prussian order Pour le Mérite (1847); officer of the Légion d’honneur (1855).

Connections

People linked to David Brewster across this collection. A name without a link has no page here yet. See David Brewster in the network of all 337 lives →

Influenced by

• Isaac Newton
• Étienne-Louis Malus

Influenced

• Gustav Kirchhoff

Colleagues and collaborators

• Charles Babbage 1830-1831
• Jean-Baptiste Biot
• John Herschel
• John Hall Gladstone 1860

Rivals

• Augustin-Jean Fresnel
• Charles Wheatstone
• Michel Chasles

Friends

• Mary Somerville
• William Henry Fox Talbot

Also mentioned in this life

• Robert Bunsen
• Johann Heinrich Lambert
• Count Rumford

Fields

• optics 38 lives

On CoursesHub

• Optics & Photonics

Sources and further reading

Biography

1. David Brewster, Wikipedia.
2. David Brewster, German Wikipedia (RSE offices and medals).
3. J. J. O’Connor and E. F. Robertson, Sir David Brewster, MacTutor History of Mathematics.
4. M. M. Gordon, The Home Life of Sir David Brewster, Edinburgh, 1869.
5. A. D. Morrison-Low and J. R. R. Christie (eds.), “Martyr of Science”: Sir David Brewster 1781–1868, Royal Scottish Museum, 1984.

Physics

1. D. Brewster, On the laws which regulate the polarisation of light by reflexion from transparent bodies, Phil. Trans. R. Soc. 105 (1815).
2. D. Brewster, On the laws which regulate the absorption of polarised light by doubly refracting crystals, Phil. Trans. R. Soc. 109 (1819).
3. D. Brewster, A Treatise on the Kaleidoscope, Edinburgh, 1819.
4. D. Brewster, The Stereoscope: Its History, Theory, and Construction, London, 1856.
5. D. Brewster and J. H. Gladstone, On the lines of the solar spectrum, Phil. Trans. R. Soc. 150 (1860).
6. M. Born and E. Wolf, Principles of Optics, Cambridge University Press (Fresnel equations; crystal optics).
7. E. Hecht, Optics, Pearson (Brewster’s angle, piles of plates, photoelasticity).
8. Photoelasticity, Kaleidoscope, Stereoscope, Fresnel lens, Wikipedia.
9. H. S. M. Coxeter, Regular Polytopes, Dover (kaleidoscopes and reflection groups).