Joseph von Fraunhofer

How to use this course

Joseph von Fraunhofer had almost no schooling and died at 39. In about fifteen years of work he turned lens-making from a craft of trial and error into a science of measurement. To measure his glass he needed sharp reference colours, and he found them in the dark lines of sunlight. To measure the wavelengths of those lines he built the first diffraction gratings. His telescopes carried out the first reliable measurements of distances to stars. Spectroscopy, the science that tells us what stars are made of, begins in his workshop.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Part II follows Fraunhofer’s own path: from measuring glass (Lessons 04 to 06) to diffraction and gratings (Lessons 07 and 08), then to what later scientists made of his lines (Lesson 09) and to the telescopes that carried his name (Lesson 10).

Timeline

Straubing and the collapsed house, 1787–1801

Joseph Fraunhofer was born on 6 March 1787 in Straubing, a town on the Danube in Lower Bavaria. He was the eleventh child of Franz Xaver Fraunhofer, a glazier, and Maria Anna Fröhlich. Both parents came from families in the glass trade. His mother died in 1797 and his father in 1798, leaving him an orphan at eleven with very little schooling.

His guardian apprenticed him in Munich to Philipp Anton Weichselberger, a mirror-maker and glass-cutter who worked him hard and gave him no time to study. On 21 July 1801 two old houses in a narrow Munich lane collapsed. Fraunhofer was buried in the rubble of Weichselberger’s building. Rescuers worked for about four hours, and he was the only occupant brought out alive. The master’s wife died.

A royal rescue

The Elector of Bavaria, Maximilian IV Joseph, came to the site, and the rescue made the boy briefly famous. The Elector gave him money and pressed his master to allow him time to study. Another visitor was Joseph Utzschneider, a government official and entrepreneur, who lent him books on optics and mathematics. Fraunhofer used the Elector’s gift to buy a glass-grinding machine and to free himself from part of his apprenticeship. Within a few years Utzschneider would bring him into the most advanced optical business in Germany.

Benediktbeuern and the art of glass, 1806–1814

In the first years of the century Utzschneider, the instrument maker Georg Reichenbach and the mechanic Joseph Liebherr set up the Mathematisch-Mechanisches Institut in Munich to make surveying and astronomical instruments. Such instruments needed lenses, and good optical glass was then a near-monopoly of a few English makers. The firm set up its own glassworks. In 1806 Fraunhofer joined its optical workshop, and in 1807 the workshop moved to Benediktbeuern, a former Benedictine monastery in the Alpine foothills that the state had secularized.

The glass melting was in the hands of Pierre-Louis Guinand, a Swiss craftsman who had discovered that stirring molten glass with a clay rod removed the streaks (striae) that spoil large lenses. Fraunhofer learned the craft from him and then improved it. By 1809 he directed the optical workshop, and in 1811 he designed a new furnace. He made both crown glass, which disperses colours weakly, and flint glass, which contains lead and disperses them strongly.

From craft to calculation

Achromatic lenses, which combine crown and flint to cancel colour fringes, had been made in England since the 1750s, notably by John Dollond. Makers ground a lens, tested it, and corrected it by hand. Fraunhofer wanted to compute the lens from measured properties of the glass and then make it exactly. That required knowing the refractive index of each melt at several precisely defined colours. Colours, however, blend into one another in a spectrum. He needed fixed marks.

The dark lines, 1814–1817

Fraunhofer first found a fixed mark in artificial light. In the spectrum of a lamp flame he saw a bright, sharp double line in the yellow, always at the same place. He used it to measure indices. Then he turned his apparatus to sunlight, admitted through a narrow slit and viewed through a prism mounted on a theodolite, a surveying instrument that measured angles precisely. He saw a great many dark lines crossing the spectrum.

The English chemist William Hyde Wollaston had seen a few such lines in 1802 and taken them for boundaries between colours. Fraunhofer showed they were something else. They kept the same positions with different prisms and different glasses, they appeared in light reflected from the Moon and planets, and they were sharp. He counted 574 of them, drew a map of the spectrum, and labelled the strongest with capital letters, from A in the deep red to H in the violet. The bright yellow double line of the lamp coincided with the dark double line he called D.

The lines solved his practical problem: they gave him reference wavelengths that anyone could find again. His memoir, Bestimmung des Brechungs- und Farbenzerstreuungs-Vermögens verschiedener Glasarten, in Bezug auf die Vervollkommnung achromatischer Fernröhre (Determination of the refractive and dispersive power of different kinds of glass, with regard to the perfection of achromatic telescopes), appeared in the Denkschriften of the Bavarian Academy for 1814–15. It tabulated refractive indices of his glasses at the lines B to H. He also pointed a telescope and prism at bright stars, and found that Sirius and other stars showed lines different from the Sun’s. In 1814 he became a partner in the firm, and in 1817 the Academy elected him a corresponding member.

Refraction, dispersion and the prism

For the general reader. Glass slows light down, and it slows blue light slightly more than red. A prism therefore bends blue light more and spreads white light into a spectrum. To design lenses, Fraunhofer needed to know exactly how much each kind of glass bends each colour. He measured it by finding the angle at which a prism bends light least.

Minimum deviation

A prism of apex angle \(A\) and refractive index \(n\) deviates a ray by \(\delta=\theta_1+\theta_4-A\), where \(\theta_1\) is the angle of incidence on the first face and \(\theta_4\) the angle of emergence from the second, and the internal angles satisfy \(\sin\theta_1=n\sin\theta_2\), \(\sin\theta_4=n\sin\theta_3\), \(\theta_2+\theta_3=A\).

Proof sketch. By reversibility of light paths, if a minimum occurs at \((\theta_2,\theta_3)\) it also occurs at \((\theta_3,\theta_2)\). Differentiating \(\delta\) with \(d\theta_2=-d\theta_3\) gives \(d\delta/d\theta_2=d\theta_1/d\theta_2-d\theta_4/d\theta_3\), and \(d\theta_1/d\theta_2=n\cos\theta_2/\cos\theta_1\) is strictly increasing in \(\theta_2\) when \(n>1\). So \(d\delta/d\theta_2\) vanishes only at \(\theta_2=\theta_3\), and that point is the minimum. Substituting \(\theta_2=A/2\), \(\theta_1=(A+\delta_{\min})/2\) into Snell’s law gives the formula. The minimum is stationary, so small errors in setting the prism do not affect the measured angle to first order. This is why Fraunhofer, and every refractometer since, used it.

Dispersion

The index varies with wavelength. In transparent glass in the visible, \(n\) decreases as \(\lambda\) increases (normal dispersion), and Augustin-Louis Cauchy’s empirical formula, \(n(\lambda)\approx A_0+B_0/\lambda^2\), fits well. Opticians still describe glass by its index at a yellow line and by its dispersion between the hydrogen lines that Fraunhofer named C (656.3 nm) and F (486.1 nm). Ernst Abbe later combined these into the Abbe number

large for low-dispersion crown glasses (about 60) and small for dense flints (about 30 to 40). Today the yellow helium d line (587.6 nm) is often used in place of D; the difference is small.

The achromatic lens

For the general reader. A single glass lens brings blue light to a focus closer than red, so stars seen through it are fringed with colour. Pairing a strong converging lens of crown glass with a weaker diverging lens of flint glass can cancel the colour while keeping the focusing power. Getting the balance right needs precise measurements of both glasses, which is exactly what Fraunhofer’s dark lines provided.

The doublet condition

A thin lens of index \(n\) has power \(\phi=1/f=(n-1)K\), where \(K=1/R_1-1/R_2\) depends only on the shape. Two thin lenses in contact have power \(\phi=\phi_1+\phi_2\). Varying the wavelength from F to C changes each power by \(\delta\phi_i=(n_{F,i}-n_{C,i})K_i=\phi_i/V_i\), where \(\phi_i\) is evaluated at D.

The two glasses must have different Abbe numbers, and the larger the difference, the weaker the individual elements and the smaller their aberrations. For a converging doublet with crown first (\(V_1>V_2\)), the crown element is positive and the flint negative.

Secondary spectrum

Bringing C and F to a common focus does not bring every colour there. The residual depends on the partial dispersion \(P=(n_D-n_C)/(n_F-n_C)\) of each glass, and the focal shift of the yellow light relative to the common C–F focus is proportional to \((P_1-P_2)/(V_1-V_2)\). Ordinary crown and flint have similar \(P\) but not equal, so a small coloured halo remains. Fraunhofer measured partial dispersions at many lines and chose glasses and curvatures to minimize the visible residual; he also corrected spherical aberration and coma by his choice of curvatures. His objectives were the best of their day.

The prism spectroscope and its resolution

For the general reader. Fraunhofer saw hundreds of lines because his instrument was sharper than anything before it. The trick was to send light through a narrow slit and then to look at the spectrum with a telescope. How many lines a spectroscope can separate depends, surprisingly, only on the thickness of glass the light crosses at the prism’s base and on how strongly that glass disperses.

The arrangement

Light from a narrow slit falls on a prism, and a telescope behind it focuses each wavelength to a separate image of the slit. A dark line is the image of the slit at a wavelength where light is missing. Wollaston had used a slit with the naked eye; Fraunhofer added the telescope and mounted it on a theodolite so that each line’s angular position could be read to seconds of arc. Later instruments added a collimating lens before the prism, so the prism receives a parallel beam.

Resolving power

The resolving power is \(R=\lambda/\Delta\lambda\), where \(\Delta\lambda\) is the smallest wavelength difference that can be distinguished. By Rayleigh’s criterion, two lines are just resolved when the diffraction maximum of one falls on the first minimum of the other. A beam of width \(a\) leaving the prism has angular diffraction half-width \(\lambda/a\) (Lesson 07).

Derivation by Fermat’s principle. The emerging wavefront is a plane. Two wavelengths whose wavefronts differ in tilt by one diffraction width \(\lambda/a\) are just resolved. The ray along the base crosses glass of length \(b\); the ray past the apex crosses none. A change \(\Delta n\) changes the optical path difference between them by \(b\,\Delta n\), and tilts the outgoing wavefront by \(b\,\Delta n/a\). Setting \(b\,\Delta n/a=\lambda/a\) gives \(\Delta\lambda=\lambda/(b\,|dn/d\lambda|)\).

Fraunhofer diffraction

For the general reader. Light passing through a small opening spreads out, and the pattern far away has bright and dark bands. Fraunhofer studied this using a telescope focused at infinity, which brings the far-away pattern up close. The pattern seen this way now bears his name. Mathematically, it is the Fourier transform of the opening.

From Huygens–Fresnel to the Fourier transform

Let a plane wave of wavelength \(\lambda\), \(k=2\pi/\lambda\), fall normally on a screen with aperture function \(t(x,y)\) (1 in the opening, 0 elsewhere). By the Huygens–Fresnel principle the field at a distant point is a sum of secondary wavelets \(e^{ikr}/r\) from each point of the aperture. At distance \(L\), with observation point \((X,Y)\),

If the aperture has size \(a\) with \(a^2\ll\lambda L\), the last term is negligible.

When \(N_F\) is not small, the quadratic term matters and one obtains Fresnel diffraction, which Augustin Fresnel developed in 1818. A lens of focal length \(f\) adds exactly the quadratic phase needed to cancel it, so the pattern in its focal plane is the Fraunhofer pattern with \(\sin\theta=X/f\). Fraunhofer’s telescope did precisely this.

Single slit

For a slit of width \(a\), \(\int_{-a/2}^{a/2}e^{-ikx\sin\theta}dx=a\,\dfrac{\sin\beta}{\beta}\) with \(\beta=\tfrac12ka\sin\theta\), so

Zeros occur at \(a\sin\theta=m\lambda\), \(m=\pm1,\pm2,\dots\). The central maximum has angular half-width \(\lambda/a\): the narrower the slit, the wider the spread.

Circular aperture

For a circular opening of diameter \(D\), the transform involves the Bessel function \(J_1\). George Airy worked it out in 1835:

The first zero of \(J_1\) is at \(u\approx3.8317\), so the first dark ring is at \(\sin\theta\approx1.22\,\lambda/D\). This bright central spot, the Airy disc, limits the sharpness of every telescope (Lesson 10).

The diffraction grating

For the general reader. A prism spreads light into a spectrum, but how far it bends each colour depends on the glass. Fraunhofer wanted an absolute measure. He made a “grating” of many fine, evenly spaced wires, later of lines ruled on glass. Light passing through many slits interferes so that each wavelength leaves at a definite angle fixed only by the spacing. From the angle and the spacing he could compute the wavelength itself.

Fraunhofer’s gratings

In his 1821 memoir, Neue Modification des Lichtes durch gegenseitige Einwirkung und Beugung der Strahlen, und Gesetze derselben, Fraunhofer described gratings made of fine parallel wires wound on two screws, with spacings of order a tenth of a millimetre. Viewing a slit through such a grating with his theodolite, he saw sharp spectra on both sides, repeated at regular intervals, crossed by the same dark lines as the prism spectrum. He deduced the grating law, including oblique incidence, and measured the wavelengths of the main solar lines. He then ruled finer gratings on glass with a diamond, and on gold films, and studied gratings used in reflection. Thomas Young had found wavelengths from two-slit fringes in 1802; Fraunhofer was the first to measure the wavelengths of specific, sharp spectral lines.

The \(N\)-slit pattern

Take \(N\) slits of width \(a\) with spacing \(d\). By the convolution argument of Exercise 7.3, the amplitude is the single-slit factor times a geometric sum over the slits. With phase difference \(\varphi=kd\sin\theta\) between neighbours,

Between neighbouring principal maxima there are \(N-1\) zeros. The first zero next to order \(m\) lies where \(N\varphi/2=\pi(mN+1)\), so the principal maxima have angular half-width \(\Delta\theta=\lambda/(Nd\cos\theta_m)\). They become sharper as \(N\) grows, and the angle does not depend on the material.

Derivation. Differentiating the grating equation, the angular dispersion is \(d\theta/d\lambda=m/(d\cos\theta_m)\). Two wavelengths are resolved when their maxima differ by at least the half-width: \(m\,\Delta\lambda/(d\cos\theta_m)\ge\lambda/(Nd\cos\theta_m)\), so \(\lambda/\Delta\lambda\le mN\).

From dark lines to atoms

For the general reader. Fraunhofer did not know what made the dark lines. Forty-five years later Gustav Kirchhoff and Robert Bunsen showed that each element absorbs exactly the colours it emits when hot. The dark lines are fingerprints of elements in the Sun’s atmosphere. A further half century later, Niels Bohr explained why the hydrogen lines that Fraunhofer had called C and F sit where they do.

Kirchhoff and Bunsen, 1859

In 1859 Kirchhoff, working with Bunsen in Heidelberg, passed sunlight through a sodium flame. The dark D lines became darker. He concluded that a gas absorbs the same wavelengths it emits, and that the D lines are sodium in the solar atmosphere.

Kirchhoff’s spectroscopic rules follow. A hot dense body gives a continuous spectrum. A hot thin gas gives bright lines. A cooler gas in front of a hotter continuous source absorbs at those same wavelengths and gives dark lines. The Sun’s deep layers act as the continuous source and its cooler outer layers as the absorbing gas.

Identifications

Fraunhofer’s main lines are now identified: A and B are absorption by oxygen in the Earth’s atmosphere (telluric lines); C and F are hydrogen Hα and Hβ; D is the sodium doublet; E and part of G are iron; the b group is mostly magnesium; H and K, at 396.8 and 393.4 nm, are ionized calcium.

The hydrogen lines

In 1885 Johann Balmer found an empirical formula for the visible hydrogen lines, generalized by Johannes Rydberg:

Bohr’s model of 1913 derived it: an electron falling from level \(n\) to level 2 emits a photon of energy \(E_n-E_2\), with \(E_n=-13.6\ \mathrm{eV}/n^2\) (corrected for the finite proton mass). Absorption runs the other way, so cooler hydrogen in the solar atmosphere removes exactly these wavelengths.

Telescopes, resolution and parallax

For the general reader. Fraunhofer’s glass and lens designs made refracting telescopes of unprecedented size and quality. Mounted so that a clock could follow the stars, they let astronomers measure tiny angles. With one of them Wilhelm Struve measured thousands of double stars, and with another Friedrich Bessel measured, in 1838, the first reliable distance to a star.

Resolution

A star is effectively a point at infinity. A perfect objective of diameter \(D\) images it as the Airy pattern of Lesson 07.

For a 24 cm objective in visible light (550 nm), \(\theta_R\approx2.8\times10^{-6}\) rad \(\approx0.58''\). Struve’s Dorpat refractor, delivered in 1824, had an objective of about 24 cm, the largest refractor in the world at the time. Fraunhofer mounted it on an equatorial mount with one axis parallel to the Earth’s axis, driven by clockwork, so that a star stayed in the field while the observer measured it. With it Struve measured 2,714 double and multiple stars between 1824 and 1837.

Parallax

As the Earth orbits the Sun, a nearby star shifts against distant ones. Its annual parallax \(p\), half the full shift, is related to its distance by

Every known parallax is below one second of arc. Bessel, at Königsberg, used a heliometer of about 16 cm aperture built by Fraunhofer’s firm and delivered after his death. A heliometer’s objective is cut in two along a diameter, and the halves slide along the cut, each forming its own image. Sliding them until the image of one star from one half falls on another star from the other half measures their separation by a micrometer screw, with high precision. In 1838 Bessel announced a parallax of 0.314″ for 61 Cygni. Struve announced a parallax for Vega in 1837 and confirmed it in 1840, and Thomas Henderson measured Alpha Centauri at about the same time.

Precision beyond resolution

Bessel’s parallax was smaller than the resolution limit of his instrument. There is no contradiction. Resolution limits the ability to separate two images; position measurement locates the centre of a single image. With \(N\) independent measurements, or \(N\) detected photons, the centroid of an image of width \(\sigma\) is found to about \(\sigma/\sqrt N\).

Munich, fame and early death, 1819–1826

In 1819 the optical workshop returned from Benediktbeuern to Munich, while the glassworks stayed in the monastery. Fraunhofer, now a partner, ran the firm that became known as Utzschneider und Fraunhofer. Its telescopes, microscopes, heliometers and surveying instruments went to observatories across Europe, and its objective lenses were the best available. He kept his methods for making glass secret, and the firm’s reputation rested on this monopoly.

He also became a scientist in his own right. His 1821 memoir on diffraction appeared in the Academy’s Denkschriften, and a shorter report on further experiments, including gratings used in reflection, followed in 1823. The Bavarian Academy made him an extraordinary member in 1821. The University of Erlangen gave him an honorary doctorate in 1822. In 1823 he became conservator of the Academy’s physical cabinet, with a professor’s salary, and in the summer of that year lectured on mathematical and physical optics. In 1824 King Maximilian I Joseph made him a knight of the Order of Merit of the Bavarian Crown, which conferred personal nobility and the name von Fraunhofer. The same year he was elected to the Leopoldina, and the great refractor arrived at Dorpat.

Illness and death

His health had been poor for years. In 1825 he caught a severe chill on a raft trip on the Isar and did not recover. He died in Munich on 7 June 1826, aged 39, of a chest illness. It has been suggested that long exposure to heavy-metal vapours from the glass furnaces contributed. He never married. He was buried in Munich’s Old Southern Cemetery, and his memorial carries the Latin phrase Approximavit sidera, “he brought the stars closer”.

The stars brought closer

After his death the firm continued under his successors, including Georg Merz, and kept supplying great refractors. A 9-inch refractor of Fraunhofer’s design at the new Berlin Observatory was the telescope with which Johann Galle found Neptune in 1846. Bessel’s heliometer measurement of 1838 gave the first reliable distance to a star.

The lines he mapped became the foundation of astrophysics. Kirchhoff and Bunsen’s explanation in 1859 turned them into a chemical analysis of the Sun and stars. Within a decade William Huggins and Angelo Secchi were classifying stellar spectra, and the Doppler shifts of the same lines soon gave stellar velocities. The grating, refined by Henry Rowland in the 1880s, became the standard tool of spectroscopy, from the laboratory to space telescopes.

Legacy

• Optics: computed achromatic objectives, the use of spectral lines as wavelength standards, and the letters C, D, F still used to specify optical glass.
• Physics: the diffraction grating, the measurement of wavelengths, and the far-field limit of diffraction, called Fraunhofer diffraction.
• Astronomy: stellar spectroscopy, the clock-driven equatorial mount, and the instruments of the first parallax measurements.
• Honours: Germany’s largest organization for applied research, the Fraunhofer-Gesellschaft, founded in 1949, bears his name, as does a lunar crater. The former glassworks at Benediktbeuern is preserved as a museum.

Sources and further reading

Biography

1. Joseph von Fraunhofer, Wikipedia.
2. Fraunhofer, Joseph von, Neue Deutsche Biographie 5 (1961) and Allgemeine Deutsche Biographie, Deutsche Biographie.
3. Joseph von Fraunhofer, New World Encyclopedia.
4. M. W. Jackson, Spectrum of Belief: Joseph von Fraunhofer and the Craft of Precision Optics, MIT Press, 2000.
5. Deutsches Museum Archive, Nachlass Fraunhofer (NL 014), finding aid.

Physics

1. J. Fraunhofer, Bestimmung des Brechungs- und Farbenzerstreuungs-Vermögens verschiedener Glasarten, in Bezug auf die Vervollkommnung achromatischer Fernröhre, Denkschriften der Königlichen Akademie der Wissenschaften zu München 5 (1814–15).
2. J. Fraunhofer, Neue Modification des Lichtes durch gegenseitige Einwirkung und Beugung der Strahlen, und Gesetze derselben, Denkschriften 8 (1821–22).
3. E. Lommel (ed.), Joseph von Fraunhofer’s gesammelte Schriften, Munich, 1888.
4. Fraunhofer lines, Wikipedia (modern wavelengths and identifications).
5. M. Born and E. Wolf, Principles of Optics, Cambridge University Press (diffraction, gratings, resolving power).
6. E. Hecht, Optics, Pearson (prisms, achromats, Fraunhofer diffraction).
7. G. Kirchhoff, Über die Fraunhofer’schen Linien, Monatsberichte der Königlichen Preussischen Akademie der Wissenschaften zu Berlin (1859).
8. M. Perryman, The history of astrometry, arXiv:1209.3563.