How to use this course
Hippolyte Fizeau never held a teaching chair and never needed one. His father’s fortune let him work as a private experimenter, and in a few years around 1850 he produced a sequence of experiments that set the agenda of optics for half a century. He measured the speed of light between two Paris hills, compared it in air and water, measured it in flowing water, and showed how the motion of a star should displace the lines in its spectrum. Later he turned the wavelength of light into a measuring rod and proposed the method by which the sizes of stars were eventually measured.
Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Numbers are given in modern SI units. Where Fizeau’s own figures are known, they are quoted and checked against the formulas.
Timeline
| 1819 | Born 23 September in Paris, eldest son of Louis Aimé Fizeau, a physician, and Béatrice Fizeau. |
| 1823 | His father becomes professor of internal pathology at the Paris Faculty of Medicine. |
| 1839 | September: attends Louis Daguerre’s public course on the daguerreotype, with Léon Foucault. |
| 1840 | Enters the Paris medical school; severe migraines make him give it up. First note on improving daguerreotypes with a gold salt. |
| 1841 | Uses bromine to shorten exposure times. |
| 1844 | Patent for a method of engraving printing plates from daguerreotypes. |
| 1845 | With Foucault, at François Arago’s suggestion, takes the first detailed photographs of the Sun. |
| 1847 | With Foucault, studies the interference of radiant heat. |
| 1848 | December: explains in a Paris lecture how the motion of a star should shift the lines of its spectrum. |
| 1849 | July: toothed-wheel measurement of the speed of light between Suresnes and Montmartre. Knight of the Légion d’honneur. |
| 1850 | With Louis Breguet, confirms that light travels more slowly in water than in air, about seven weeks after Foucault. |
| 1851 | The moving-water experiment, presented to the Academy on 29 September. Stands unsuccessfully for the Academy in March. |
| 1853 | Marries Thérèse Valentine de Jussieu, daughter of the botanist Adrien de Jussieu. Proposes adding a condenser to the induction coil. |
| 1856 | Awarded the triennial Grand Prix of the Institut de France. |
| 1860 | Elected to the Académie des sciences, physics section. |
| 1863 | Death of his wife. He withdraws increasingly to Venteuil, near Jouarre. |
| 1864 | Interference with sodium light; proposes the wavelength of light as a standard of length. Begins a long series of measurements of thermal expansion. |
| 1866 | Rumford Medal of the Royal Society of London. |
| 1868 | Proposes measuring the angular diameters of stars by interference. |
| 1873 | Édouard Stephan tries the method at Marseille. |
| 1874 | Helps plan the French photographic observations of the transit of Venus. |
| 1875 | Foreign Member of the Royal Society; officer of the Légion d’honneur. |
| 1877–78 | Vice-president, then president, of the Académie des sciences. Member of the Bureau des longitudes from 1878. |
| 1889 | His name is among the 72 inscribed on the Eiffel Tower; he is one of very few of them still alive. |
| 1896 | Dies 18 September at Venteuil, five days before his 77th birthday. |
A family of physicians, 1819–1840
Armand Hippolyte Louis Fizeau was born in Paris on 23 September 1819, the eldest son of Louis Aimé Fizeau and Béatrice Fizeau. The family had produced doctors for several generations. His father became professor of internal pathology at the Paris Faculty of Medicine in 1823 and built a substantial fortune, which later freed his son from any need to earn a living.
Fizeau was educated at the Collège Stanislas in Paris. MacTutor records that Léon Foucault, born five days before him in September 1819, was a friend from this period; the two would work as partners and then as rivals. The family also kept a house at Suresnes, on the slopes below Mont Valérien west of Paris. Its small turret would serve as one end of the most famous experiment of his life.
Around 1840 he entered his father’s medical school. Severe migraines forced him to abandon the course. He travelled to recover his health and, when he returned, turned to physics. He had no formal training in it. He learned from his brother’s lecture notes from the École Polytechnique, attended Henri Victor Regnault’s optics course at the Collège de France, and went to Arago’s public lectures on astronomy at the Paris Observatory.
The daguerreotype years, 1839–1845
In August 1839 the French state made Louis Daguerre’s photographic process public, and in September Daguerre gave free demonstrations in Paris. Fizeau and Foucault attended. The early daguerreotype was slow, fragile and faint: a silvered copper plate was sensitized with iodine vapour, exposed for many minutes, developed in mercury vapour and fixed.
Fizeau made three improvements. In 1840 he proposed toning the finished image with a gold salt, which made it brighter and much more durable; gold toning became standard practice. In 1841 he described the use of bromine as an accelerating agent, which helped bring exposure times down from many minutes to well under a minute. From 1843 he developed a method of etching the daguerreotype plate itself so that it could serve as a printing plate, and patented it in 1844. The engraving method had little commercial future, because paper processes soon displaced the daguerreotype.
The photographic work brought him into Arago’s orbit. Arago, director of the Paris Observatory and permanent secretary of the Academy of Sciences, had championed Daguerre’s invention before the Chamber of Deputies. In 1845 he suggested that Fizeau and Foucault photograph the image of the Sun formed by a telescope. Their daguerreotype of 1845 is the first detailed photograph of the Sun. It shows groups of sunspots and a clear darkening of the disc towards its edge. Limb darkening means that one sees deeper, hotter layers at the centre of the disc than at the edge, so the light comes from a gas whose temperature falls outwards, not from a solid or liquid surface (Exercise 4.3).
Arago’s circle and Foucault, 1845–1850
Arago was the elder statesman of the wave theory of light. In the 1810s and 1820s he had worked with Augustin-Jean Fresnel on interference and polarization, and he had proposed several experiments to decide between the wave and particle theories. Fizeau and Foucault became, in effect, his experimental heirs. In 1847 they showed that radiant heat, the invisible radiation beyond the red end of the spectrum, forms interference fringes just as visible light does (Lesson 04). They also studied the interference of light over long path differences and chromatic polarization.
Their partnership ended around 1849 or 1850. The two had different temperaments, and both were now pursuing the same prize: the measurement of the speed of light, and the comparison of its speed in air and water that Arago had long advocated. A sealed letter deposited by Foucault with the Academy on 27 May 1850, and opened only in 1983, shows that they had nevertheless worked together that spring on a test for the dragging of light by moving air. They found no effect.
In December 1848 Fizeau gave a lecture in Paris on the effect of motion on the pitch of sound and the colour of light. He did not know that Christian Doppler had published the principle for both in Prague six years earlier. Fizeau’s contribution was to see what an astronomer could actually measure (Lesson 05). The next year he made the measurement that made his name, using a toothed wheel in the turret of the family house at Suresnes and a mirror on Montmartre (Lesson 06). He was made a knight of the Légion d’honneur in 1849, aged 29.
Interference of light and radiant heat
For the general reader. When two waves overlap, crests can reinforce crests or cancel troughs, producing a pattern of bright and dark bands. Thomas Young and Fresnel had used such patterns to prove that light is a wave. Fizeau and Foucault showed in 1847 that the invisible “heat rays” beyond the red end of the spectrum also make such bands. Heat radiation and light are the same kind of wave, differing only in wavelength.
- Derive the two-beam interference formula and the fringe spacing.
- Understand why white-light fringes are few and coloured.
- Relate limb darkening to the temperature structure of the Sun.
Two-beam interference
Represent two monochromatic waves at a point by complex amplitudes \(E_1=a_1e^{i\phi_1}\) and \(E_2=a_2e^{i\phi_2}\) with the same frequency. The detected intensity is the time average of the squared field, proportional to \(|E_1+E_2|^2\).
If two mutually coherent beams of intensities \(I_1,I_2\) and wavelength \(\lambda\) meet with optical path difference \(\Delta\), the intensity is
\[ I = I_1+I_2+2\sqrt{I_1I_2}\,\cos\!\Big(\frac{2\pi\Delta}{\lambda}\Big). \]For two slits a distance \(d\) apart viewed on a screen at distance \(D\gg d\), \(\Delta\approx d\,y/D\) at height \(y\), and the bright fringes are spaced by \(\delta y=\lambda D/d\).
The proof is one line: \(|E_1+E_2|^2=a_1^2+a_2^2+2a_1a_2\cos(\phi_1-\phi_2)\), with \(\phi_1-\phi_2=2\pi\Delta/\lambda\). The fringe visibility \(V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})=2\sqrt{I_1I_2}/(I_1+I_2)\) equals 1 when the beams are equally bright.
Heat rays
William Herschel had found in 1800 that a thermometer placed beyond the red end of a solar spectrum still warms. Whether this radiation was light of another colour or a separate agent was disputed. Interference decides the question, because only a wave produces fringes, and the fringe spacing measures the wavelength. The fringe formula shows that radiation of longer wavelength gives proportionally wider fringes, so a sensitive heat detector moved across the pattern should find maxima and minima on a coarser scale than the visible ones. That is what Fizeau and Foucault reported.
White light
With white light each wavelength produces its own fringe system. All systems agree at \(\Delta=0\), so the central fringe is white. Away from the centre the systems drift out of step, and after a few fringes the colours blend into uniform illumination. Fizeau and Foucault studied how far fringes remain visible as the path difference grows, which is a measurement of the spectral width of the light. Lesson 09 makes this precise.
Exercise 4.1Two slits 0.50 mm apart are viewed on a screen 2.0 m away. Find the fringe spacing for green light (\(\lambda=550\) nm) and for near-infrared radiation of wavelength 1.5 µm.
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Exercise 4.2Visible light spans roughly 400–700 nm. At what order \(m\) does the \(m\)-th bright fringe of 700 nm light fall on the \((m+1)\)-th bright fringe of 400 nm light? What does this say about the number of white-light fringes one can see?
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Exercise 4.3A standard approximation to solar limb darkening is \(I(\mu)=I(1)\,\tfrac{2+3\mu}{5}\), where \(\mu=\cos\theta\) and \(\theta\) is the angle between the line of sight and the local vertical. Find the brightness at the limb relative to the centre, and explain in one sentence why a solid incandescent surface would not show this effect.
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The Doppler–Fizeau effect
For the general reader. The pitch of a passing siren falls as it goes by. Christian Doppler proposed in 1842 that light should behave the same way, and suggested that the colours of stars come from their motion. Fizeau saw in 1848 why that cannot work and what can. Motion shifts the whole spectrum, so the overall colour hardly changes, but the dark lines in the spectrum move, and their displacement measures the speed along the line of sight. In France the effect is still called the Doppler–Fizeau effect.
- Derive the Doppler formulas for waves in a medium and for light.
- See why line displacements, not colours, reveal stellar motion.
- Estimate the shifts produced by the Earth’s orbital motion and by planets around other stars.
Waves in a medium
Let waves travel at speed \(w\) through a still medium. A source emitting frequency \(f\) moves towards an observer at speed \(v_s\), and the observer moves towards the source at speed \(v_o\), both relative to the medium. Successive crests are emitted \(1/f\) apart in time, during which the source advances \(v_s/f\), so the wavelength in the medium is \(\lambda'=(w-v_s)/f\). The observer meets crests at relative speed \(w+v_o\).
To first order in \(v/w\) this depends only on the relative velocity: \(f_{\text{obs}}\approx f\,(1+v_r/w)\), where \(v_r=v_o+v_s\) is the speed of approach.
Light
For light in vacuum there is no medium, and only the relative velocity can matter. The exact formula follows from the Lorentz transformation of the wave four-vector \((\omega/c,\mathbf k)\).
If source and observer approach each other along the line joining them with relative speed \(v=\beta c\),
\[ f_{\text{obs}} = f\sqrt{\frac{1+\beta}{1-\beta}}, \qquad \frac{\Delta\lambda}{\lambda}=-\beta+O(\beta^2). \]For recession replace \(\beta\) by \(-\beta\). For motion at angle \(\theta\) to the line of sight in the observer’s frame, \(f_{\text{obs}}=f/\big(\gamma(1-\beta\cos\theta)\big)\) with \(\gamma=(1-\beta^2)^{-1/2}\).
Derivation. In the source frame \(S'\) the phase of the wave is \(\omega t'-k x'\) with \(\omega=ck\). Phase is invariant. Substituting the Lorentz transformation \(t'=\gamma(t-\beta x/c)\), \(x'=\gamma(x-\beta ct)\) turns \(t'-x'/c\) into \(\gamma(1+\beta)(t-x/c)\). Here \(S'\) moves in the \(+x\) direction relative to the observer’s frame \(S\), so a source at rest in \(S'\) approaches an observer further along the \(x\)-axis, and the observed frequency is \(\omega_{\text{obs}}=\gamma(1+\beta)\omega\). Since \(\gamma(1+\beta)=\sqrt{(1+\beta)/(1-\beta)}\), the formula follows.
Fizeau’s point
Doppler had argued that the colours of double stars come from their orbital motion. The spectra of stars are broad, extending well beyond the visible on both sides. A shift moves some invisible light into view at one end and some visible light out at the other, and for any plausible stellar speed the net change of colour is imperceptible. Fizeau observed that the dark lines discovered by Fraunhofer are sharp markers. Their displacement relative to the same lines produced in a laboratory measures \(\Delta\lambda/\lambda\) directly, and he noted that the method could detect even the motion of the observer. Stellar radial velocities measured this way became one of the main tools of astrophysics.
Exercise 5.1The Earth’s orbital speed is 29.8 km/s. What is the largest annual shift of the sodium D line at 589.0 nm seen in the spectrum of a star in the plane of the Earth’s orbit? Compare it with the separation of the two D lines, 0.6 nm.
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Exercise 5.2Doppler wanted motion to change a star’s colour. Estimate the speed needed to shift a line from 550 nm (green) to 600 nm (yellow-orange). What would Fizeau have said?
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Exercise 5.3The planet 51 Pegasi b makes its star wobble with a line-of-sight velocity amplitude of about 56 m/s. Find the fractional wavelength shift and the shift of a line at 500 nm.
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Exercise 5.4Show that the relativistic formula reduces to the classical one when \(\beta\ll1\), to first order, whether the source or the observer moves.
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The toothed wheel, 1849
For the general reader. Before 1849 the speed of light had been measured only from astronomy: by Ole Rømer from the eclipses of Jupiter’s moons and by James Bradley from the aberration of starlight. Fizeau measured it on the ground. He sent light through a gap between the teeth of a spinning wheel to a mirror about 8.6 km away and back. When the wheel spun fast enough, the returning light met a tooth instead of a gap and the image went dark. From the speed of the wheel and the distance, he found the speed of light.
- Derive the extinction condition and compute Fizeau’s value.
- Understand the optics that made a light beam usable over 8.6 km.
- Estimate the main sources of error.
The apparatus
The transmitting station was the turret of the family house at Suresnes; the reflector stood on Montmartre, 8633 m away. A bright source was imaged by a lens, via a partly reflecting glass plate, onto the rim of the toothed wheel. A telescope objective turned the light leaving each gap into a parallel beam. At Montmartre a second objective focused the beam onto a mirror placed at its focus, which sent it back along the same path. The returning light was brought to a focus at the same point of the wheel rim and viewed through an eyepiece behind the glass plate. The wheel had 720 teeth, with gaps equal in width to the teeth, and was driven by a clockwork with a revolution counter.
Let the wheel have \(N\) teeth and gaps of equal width, turn at \(\nu\) revolutions per second, and let the mirror be at distance \(D\). The returning light is eclipsed when the round-trip time equals an odd number of half tooth periods:
\[ \frac{2D}{c}=\frac{2k-1}{2N\nu_k},\qquad\text{so}\qquad c=\frac{4ND\nu_k}{2k-1},\quad k=1,2,\dots \]Maximum brightness returns at \(\nu=2\nu_1,4\nu_1,\dots\), when the wheel advances by a whole number of tooth periods during the round trip.
One tooth period is \(1/(N\nu)\) seconds; a gap is replaced by a tooth after half of it. Fizeau observed the first extinction at about 12.6 revolutions per second and the return of the light at about 25.2. Then
\[ c = 4\times720\times8633\ \text{m}\times12.6\ \text{s}^{-1}\approx3.13\times10^{8}\ \text{m/s}. \]Fizeau averaged many observations and published a value of about 315 000 km/s, roughly 5% above the modern value \(c=299\,792.458\) km/s. The main difficulty was judging the exact moment of extinction, because the dimming is gradual. The method was later refined on longer baselines by Alfred Cornu in the 1870s.
Exercise 6.1Using \(c=299\,792\) km/s, at what rotation rate should Fizeau’s wheel have produced the first extinction? The second? By what percentage did his reading of 12.6 rev/s differ?
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Exercise 6.2Suppose the observer can judge the extinction only to within ±0.5 rev/s at the first minimum. What is the resulting uncertainty in \(c\)? How does using the \(k\)-th extinction help, if the reading error stays ±0.5 rev/s?
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Exercise 6.3A teaching laboratory has a straight path of only 50 m. With a wheel of 720 teeth, what rotation rate gives the first extinction? Is this practical?
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Air against water, 1850
For the general reader. In the wave theory, light bends towards the perpendicular on entering water because it slows down. In Newton’s particle theory it bends because water attracts the particles and speeds them up. Arago had pointed out that a direct comparison of the speed of light in air and in water would decide between the theories. In 1850 Foucault, and Fizeau with the instrument maker Louis Breguet, did the experiment with rotating mirrors. Light was slower in water.
- Derive Snell’s law in the particle and wave theories and compare their predictions.
- Derive the deflection produced by a rotating mirror.
- Estimate the time differences the 1850 experiments had to detect.
Two derivations of Snell’s law
Let light pass from air into water with angles of incidence and refraction \(i\) and \(r\), and let \(n=\sin i/\sin r\approx1.33\). The particle theory requires \(v_{\text{water}}=n\,v_{\text{air}}\); the wave theory requires \(v_{\text{water}}=v_{\text{air}}/n\).
Particles. An attractive force normal to the surface changes only the normal component of velocity. The tangential components are equal: \(v_1\sin i=v_2\sin r\), so \(v_2/v_1=\sin i/\sin r=n\).
Waves. By Huygens’s construction, a wavefront meeting the surface advances a distance \(v_1t\) in air while the wavelet in water advances \(v_2t\), over the same stretch \(\ell\) of surface: \(\sin i=v_1t/\ell\), \(\sin r=v_2t/\ell\). Hence \(v_1/v_2=n\).
The rotating mirror
A mirror turning at angular velocity \(\Omega\) reflects a beam to a distant fixed mirror at distance \(d\). The beam returns after \(\tau=2d/v\), by which time the rotating mirror has turned by \(\Omega\tau\). Reflection doubles angles, so the returned beam is deviated from its outgoing direction by
\[ \delta = 2\Omega\tau=\frac{4\Omega d}{v}. \]If one of two otherwise identical paths passes through a tube of water, its return image is deflected more if light is slower in water, and less if it is faster. The comparison is qualitative and needs no knowledge of \(c\). Foucault reported first, in the spring of 1850; Fizeau and Breguet confirmed the result about seven weeks later. The wave theory had been accepted on the evidence of interference since Fresnel, but this was the experiment Arago had called decisive.
Exercise 7.1A tube of water 3.0 m long (\(n=1.333\)) is placed in a light path. By how much does the transit time change according to each theory?
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Exercise 7.2A mirror turns at 800 revolutions per second, and the fixed mirror is 4.0 m away in air. Find the deviation \(\delta\) and the displacement of the return image as seen 1.0 m from the rotating mirror. By what fraction does \(\delta\) change if the whole path is filled with water?
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Light in moving water, 1851
For the general reader. If light is a wave in an ether, does a moving transparent body carry the ether, and the light, along with it? Fresnel had predicted in 1818 that it does, but only partly. Fizeau sent two beams of light in opposite directions through flowing water and compared them by interference. The light was dragged, by exactly Fresnel’s fraction. Half a century later this result became one of the simplest confirmations of Einstein’s rule for adding velocities.
- State Fresnel’s drag coefficient and derive the fringe shift in Fizeau’s apparatus.
- Compare the prediction with Fizeau’s numbers and with Michelson and Morley’s repetition.
- Derive the drag coefficient from special relativity, and the dispersion correction.
Fresnel’s hypothesis
In a medium of refractive index \(n\) moving with speed \(v\) along the direction of propagation, the speed of light relative to the laboratory is
\[ u = \frac{c}{n} + f\,v,\qquad f = 1-\frac{1}{n^2}. \]Fresnel derived the coefficient from a model in which the ether density inside matter is proportional to \(n^2\) and only the excess is carried along. Its purpose was to explain why Arago’s experiments of 1810 showed no effect of the Earth’s motion on the refraction of starlight. George Gabriel Stokes proposed instead that matter drags the ether completely, \(f=1\).
The apparatus
Light from a source was split into two beams passing through two parallel tubes of water, each of length \(L\). The water flowed in opposite directions in the two tubes. A mirror at the far end returned each beam through the other tube, so that one beam travelled with the current in both tubes and the other against it in both. The beams were then recombined and the fringes observed. Because both beams followed the same path in opposite senses, temperature and pressure disturbances affected them equally.
To first order in \(v/c\), the flow shifts the fringe pattern relative to still water by
\[ d=\frac{4Ln^2 f v}{\lambda c} \]fringes, where \(\lambda\) is the vacuum wavelength. Reversing the flow doubles the observed displacement.
Proof. Each beam crosses a total length \(2L\) of water. The difference of transit times is
\[ \Delta t=\frac{2L}{c/n-fv}-\frac{2L}{c/n+fv}=\frac{4Lfv}{(c/n)^2-f^2v^2}\approx\frac{4Ln^2fv}{c^2}, \]and the number of fringes is \(c\,\Delta t/\lambda\).
Fizeau’s tubes were about 1.49 m long and the water ran at about 7.06 m/s. With \(n=1.333\) and \(\lambda\approx526\) nm, full drag predicts about 0.47 fringe and Fresnel’s coefficient about 0.21. Fizeau measured about 0.23. He presented the result on 29 September 1851 under a title announcing that the motion of bodies changes the speed of light within them. Albert Michelson and Edward Morley repeated the experiment in 1886 with a larger apparatus and found \(f=0.434\pm0.02\) against Fresnel’s 0.437 for their wavelength.
The relativistic explanation
In the rest frame of the water light travels at \(c/n\). By the relativistic addition of velocities, in the laboratory
\[ u=\frac{c/n+v}{1+v/(nc)} = \frac{c}{n}+v\Big(1-\frac{1}{n^2}\Big)+O\Big(\frac{v^2}{c}\Big). \]Fresnel’s coefficient is therefore no property of the ether. It is a consequence of the kinematics of special relativity, which Einstein had published two years before Max von Laue noticed this. Hendrik Lorentz had shown in 1895 that because the light’s frequency in the moving water is Doppler shifted, dispersion adds a further term:
\[ f = 1-\frac{1}{n^2}-\frac{\lambda}{n}\frac{dn}{d\lambda}. \]Pieter Zeeman confirmed the dispersion term in 1914–15 with light of several wavelengths.
Exercise 8.1Using \(L=1.487\) m, \(v=7.059\) m/s, \(n=1.333\), \(\lambda=526\) nm, compute the predicted shift for full drag and for Fresnel drag.
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Exercise 8.2Expand \(u=(c/n+v)/(1+v/(nc))\) to first order in \(v\) and recover Fresnel’s coefficient.
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Exercise 8.3For water near 526 nm, \(dn/d\lambda\approx-4.0\times10^{4}\ \text{m}^{-1}\). Find the dispersion-corrected drag coefficient and compare with Michelson and Morley’s measurement.
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Exercise 8.4Fizeau also ran air through the tubes and saw no shift. Why was this expected under Fresnel’s hypothesis, with \(n_{\text{air}}=1.0003\)?
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Interference as a ruler
For the general reader. A wavelength of light is about half a micrometre, and interference fringes move by one whole fringe when a gap changes by half a wavelength. From 1864 Fizeau used this to measure how solids expand when heated, with a sensitivity no mechanical gauge could approach. In 1864 he proposed that the wavelength of a natural light source should become the standard of length. It did, in 1960.
- Derive the fringe condition for a thin air film and use it to measure expansion.
- Explain how a doublet source makes fringes fade and return.
- Connect Fizeau’s proposal to the definitions of the metre.
Fringes of a thin film
Monochromatic light of wavelength \(\lambda\) falls normally on an air gap of thickness \(h\) between two glass surfaces. Light reflected from the two surfaces interferes, with a dark fringe wherever
\[ 2h = m\lambda,\qquad m=0,1,2,\dots \]If the gap at a fixed point changes by \(\Delta h\), the number of fringes that pass that point is \(N=2\Delta h/\lambda\).
The round-trip path difference is \(2h\), and reflection at the lower (air-to-glass) surface adds a phase of \(\pi\), which turns the condition for constructive interference into the condition for darkness. This is the principle of the Fizeau interferometer, still the standard instrument for testing optical flats. In Fizeau’s dilatometer, a sample stood beneath a glass plate so that its upper face and the plate bounded a thin air film. As the apparatus was heated, the fringes moved across a reference mark, and their count gave the change of the gap in half-wavelengths. Between 1864 and 1870 he measured the expansion of many crystals and other substances this way. In 1867 he found that silver iodide contracts when heated.
A doublet source
A source emits two lines of equal intensity at wavenumbers \(\sigma_1,\sigma_2\) (\(\sigma=1/\lambda\)), with \(\Delta\sigma=|\sigma_1-\sigma_2|\). In a two-beam interferometer with path difference \(x\), the fringe visibility is
\[ V(x)=|\cos(\pi\,\Delta\sigma\,x)|. \]Adding the intensities \(1+\cos(2\pi\sigma_1x)\) and \(1+\cos(2\pi\sigma_2x)\) gives \(2+2\cos(\pi\Delta\sigma x)\cos(2\pi\bar\sigma x)\): fast fringes with a slowly varying envelope. Fizeau observed exactly this behaviour with the yellow light of a sodium flame, whose D line is a doublet: the fringes faded, vanished and returned as the path difference grew. The regularity of the effect over very large fringe counts persuaded him that the wavelength of a natural source is a stable, reproducible quantity, suitable as a standard of length.
Exercise 9.1A crystal 10.0 mm long is heated by 50 K. In sodium light (\(\lambda=589\) nm) 25 fringes pass the reference mark. Find the linear expansion coefficient, assuming the change in gap equals the change in length of the crystal.
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Exercise 9.2The sodium D lines are at 588.995 nm and 589.592 nm. After how many fringes does the visibility first vanish? When does it first return to its maximum?
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Exercise 9.3In 1960 the metre was defined as 1 650 763.73 wavelengths in vacuum of an orange line of krypton-86. Find that wavelength. Since 1983 the metre has been defined by fixing \(c=299\,792\,458\) m/s exactly. What does a measurement of the speed of light now determine?
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Stellar interferometry
For the general reader. Stars are so far away that even the largest telescopes of the nineteenth century showed them only as points blurred by the air. In 1868 Fizeau suggested covering a telescope with a mask pierced by two holes. A point-like star would give sharp fringes; a star of measurable size would give fringes that fade as the holes are moved apart. The separation at which they vanish gives the star’s angular diameter. The idea took fifty years to succeed.
- Compute the fringe visibility for a double star and for a uniform disc.
- State the van Cittert–Zernike theorem.
- Follow the method from Stephan in 1873 to Michelson and Pease in 1920.
From source to visibility
Two apertures separated by a baseline \(B\) receive light from a source of small angular extent. Each point of the source at angle \(\alpha\) produces its own fringe pattern, displaced by a phase \(2\pi B\alpha/\lambda\). Different points of an incoherent source add their intensities, so the patterns average.
For an incoherent source with normalized brightness \(I(\alpha)\), \(\int I\,d\alpha=1\), observed in quasi-monochromatic light of wavelength \(\lambda\) with baseline \(B\), the complex fringe visibility is
\[ \mathcal V(B)=\int I(\alpha)\,e^{-2\pi i B\alpha/\lambda}\,d\alpha, \]the Fourier transform of the brightness at spatial frequency \(B/\lambda\). The measured visibility is \(|\mathcal V(B)|\).
(a) Two equal point sources separated by \(\alpha_0\): \(|\mathcal V|=|\cos(\pi B\alpha_0/\lambda)|\), first zero at \(B=\lambda/(2\alpha_0)\).
(b) A uniform disc of angular diameter \(\theta\): \(|\mathcal V|=\big|2J_1(x)/x\big|\) with \(x=\pi B\theta/\lambda\), first zero at \(B\approx1.22\,\lambda/\theta\).
Part (a) is immediate from two delta functions. Part (b) uses the two-dimensional Fourier transform of a disc, which is the Airy function. The key advantage of the method is that atmospheric turbulence moves the whole fringe pattern but, over small apertures, barely reduces its contrast, so the visibility can be measured even when the image itself is badly blurred.
History
Édouard Stephan put masks on the 80 cm reflector at Marseille in 1873, with apertures up to 65 cm apart. The fringes of every star he observed remained sharp, so he could only conclude that stellar diameters were much smaller than about 0.16 arcseconds. Albert Michelson measured the diameters of Jupiter’s Galilean satellites in this way at Lick Observatory in the early 1890s. In December 1920 Michelson and Francis Pease mounted a beam with movable mirrors on the 2.5 m Hooker telescope at Mount Wilson and measured the angular diameter of Betelgeuse as 0.047 arcseconds, the first measurement of the size of a star other than the Sun.
Exercise 10.1A double star has two equal components 0.50 arcsec apart. At what baseline do the fringes first vanish at \(\lambda=550\) nm?
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Exercise 10.2For Betelgeuse, \(\theta=0.047\) arcsec. At what separation of the outer mirrors should the fringes vanish at an effective wavelength of 575 nm? Why could this not be done with the bare telescope aperture?
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Exercise 10.3With Stephan’s largest baseline of 0.65 m at 500 nm, find the angular diameter of a uniform disc whose fringes would just vanish. A bright star such as Sirius has an angular diameter of about 0.006 arcsec. Why did Stephan see no effect?
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The Academy and Venteuil, 1851–1896
In March 1851 Fizeau stood for election to the Académie des sciences. In the first ballot he tied for fourth place among seven candidates and withdrew; Foucault also failed. Recognition came by another route. In 1853 he married Thérèse Valentine de Jussieu, daughter of the botanist Adrien de Jussieu and heir of the great family of French botanists. In 1856 the Institut de France awarded him the triennial Grand Prix created by Napoleon III, for his work on the speed of light, including its speed in moving bodies. On 2 January 1860 he was elected to the physics section of the Academy. Foucault followed in 1865.
His wife died in 1863. Fizeau, who had always been reserved, withdrew to his property at Venteuil near Jouarre, east of Paris, and came less often to meetings of the Academy and later of the Bureau des longitudes. He continued to experiment: on the effect of heat on the speed of light in glass (1862), on interference with sodium light (1864), on thermal expansion over the following six years, on the solar temperature, on Crookes’s radiometer, and later on magnetism. In 1853 he had proposed adding a condenser to the primary circuit of the induction coil, which greatly increased its spark, and the device became standard.
He received the Rumford Medal of the Royal Society in 1866 and was elected one of its Foreign Members in 1875, the year he became an officer of the Légion d’honneur. He helped plan the French photographic observations of the 1874 transit of Venus; Mount Fizeau on Campbell Island, south of New Zealand, where a French expedition observed the transit, is named after him. He was vice-president of the Academy in 1877 and president in 1878, and joined the Bureau des longitudes in 1878. As president he spoke on the dignity and independence of natural science and on the limits of its competence; Alfred Cornu, who wrote the notice of his scientific work, described him as a practising Christian who made no secret of his faith.
When the Eiffel Tower opened in 1889, Fizeau’s name was among the 72 scientists and engineers engraved on its frieze, and he was one of very few of them still living. He died at Venteuil on 18 September 1896, a few days before his 77th birthday.
DiscussionFizeau worked on independent means, outside teaching institutions, and published mainly short notes in the Comptes rendus. What did this freedom make possible, and what did it cost him and his field? Compare Foucault, who held only a modest post at the Observatory.
Legacy
Fizeau’s experiments were short, clean and decisive, and nearly every one founded a line of research.
- The speed of light. The toothed wheel brought the measurement of \(c\) into the laboratory. Cornu refined the method, Foucault turned the rotating mirror into a precision instrument in 1862, and Michelson pushed it to a few parts in a hundred thousand. In 1983 \(c\) became a defined constant (Exercise 9.3).
- Doppler spectroscopy. Radial velocities from line shifts measure the motions of stars, the orbits of spectroscopic binaries, the rotation of galaxies, the expansion of the universe and, since 1995, the wobble of stars with planets.
- Relativity. The moving-water experiment showed that light is only partly dragged by moving matter. Lorentz built his electron theory to explain it, and Laue showed in 1907 that special relativity explains it without an ether. Robert Shankland reported that Einstein named aberration and Fizeau’s experiment as the results that influenced him most, adding: “They were enough.”
- Interferometry. The Fizeau interferometer tests optical surfaces in every optics workshop. His proposal of a natural wavelength as a length standard was adopted in 1960. Stellar interferometry, from Michelson’s outrigger beam to today’s arrays of separate telescopes, measures the sizes and shapes of stars.
His name is on a lunar crater, on an asteroid, on streets in Paris and Suresnes, and on the Eiffel Tower. The house at Suresnes served as the town hall from 1855 to 1889.
Final projectChoose one experiment from Part II. Write a two-page account for a strong undergraduate that describes the apparatus, derives the predicted effect, compares it with Fizeau’s numbers, and explains how the same idea is used in a modern instrument.
Sources and further reading
Biography
- J. J. O’Connor and E. F. Robertson, Armand-Hippolyte-Louis Fizeau, MacTutor History of Mathematics.
- Fizeau, Armand-Hippolyte-Louis, Biographical Encyclopedia of Astronomers.
- Académie des sciences, Fonds Fizeau (64 J), archive inventory with biographical chronology.
- Hippolyte Fizeau and Hippolyte Fizeau (French), Wikipedia.
- A. Cornu, Notice sur l’œuvre scientifique de H. Fizeau, Annuaire pour l’an 1898, Bureau des longitudes.
- Armand-Hippolyte-Louis Fizeau, Catholic Encyclopedia (1909).
Physics
- H. Fizeau, Sur une expérience relative à la vitesse de propagation de la lumière, Comptes rendus 29 (1849).
- H. Fizeau, Sur les hypothèses relatives à l’éther lumineux, et sur une expérience qui paraît démontrer que le mouvement des corps change la vitesse avec laquelle la lumière se propage dans leur intérieur, Comptes rendus 33 (1851); expanded in Annales de chimie et de physique 57 (1859).
- A. A. Michelson and E. W. Morley, Influence of motion of the medium on the velocity of light, American Journal of Science 31 (1886).
- M. Laue, Die Mitführung des Lichtes durch bewegte Körper nach dem Relativitätsprinzip, Annalen der Physik 23 (1907).
- Fizeau experiment, Wikipedia.
- K. Brown, Fresnel’s drag coefficient and dispersion, MathPages (Fizeau’s and Michelson’s numbers).
- Speed of light, Wikipedia (history of measurements).
- Introduction to optical/IR interferometry: history and basic principles, arXiv:1907.07443.
- A. A. Michelson and F. G. Pease, Measurement of the diameter of α Orionis with the interferometer, Astrophysical Journal 53 (1921).
- M. Born and E. Wolf, Principles of Optics, Cambridge University Press (interference, coherence, van Cittert–Zernike theorem).