Experiment: Wave Optics
- The predictions under test
- Young: the double slit (1801–1804)
- Fresnel and the Poisson–Arago spot (1818–1819)
- The Fresnel–Arago laws and transversality
- Diffraction gratings and wavelength metrology
- Interferometry
- Stellar interferometry
- Gabor: holography and the recording of phase (1948, 1962)
- Modern repetitions and precision
This chapter is the evidence for Electromagnetic Waves and Optics. It collects the measurements that established, quantitatively and against a competing theory of equal standing, that light is a wave: Young's two-slit fringes and the first wavelength measurements [Young:1802] [Young:1804]; Fresnel's prize memoir on diffraction with the bright spot at the centre of a circular shadow that Poisson produced as a refutation and Arago found [Fresnel:1826]; the Fresnel–Arago laws, whose failure of two perpendicularly polarized beams to interfere forced the conclusion that the vibration is transverse [Arago:1819] [Fresnel:1832]; the ruled grating turned from a curiosity into the primary instrument of wavelength metrology by Fraunhofer [Fraunhofer:1821] [Fraunhofer:1823] and Rowland [Rowland:1882]; and the interferometers, from Michelson's [Michelson:1887] through his stellar beam interferometer [Michelson:1920] [Michelson:1921] to the kilometre-scale instruments of Experiment: Gravitational Waves.
Two of these experiments are shared with other chapters and are treated
here only for their optics: Fizeau's moving-water measurement
[Fizeau:1851] and the Michelson–Morley null result
[Michelson:1887] are read as interferometry here, and read as
evidence about the aether and about relativity in
Experiments: Light, the Aether, and Time. The chapter closes with the modern
repetitions, which is where the classical account runs out: the same
two-slit pattern builds up one particle at a time, for photons
[Taylor:1909] [Grangier:1986], for electrons
[Joensson:1961] [Tonomura:1989] and for molecules
[Arndt:1999] — a result that belongs to
Part VIII — The Transition to Quantum Physics and Part IX — Quantum Mechanics and is stated here
only to mark precisely where wave optics stops being the whole story.
When written out, every experiment below will use the structured
experiment environment, with apparatus, procedure, observations
(numbers and uncertainties in SI units), interpretation and primary
references given in full. The standard modern reference for the theory
being tested is [Born:1999].
Experiment: Wave Optics: all derivations of this chapter are pending.
The predictions under test
Two accounts of light
[Reserved: the state of the question about 1800. On one side the emission theory of [Newton:1704], which explains rectilinear propagation and reflection immediately, refraction by an increase of speed in the denser medium, and colour by particle species; on the other the wave account of [Huygens:1690] with its secondary wavelets, which explains refraction by a decrease of speed and gives double refraction in calcite. The prior anomaly both had to face is Grimaldi's diffraction — light bending into the geometrical shadow of a small obstacle [Grimaldi:1665] — and Newton's own rings [Newton:1704], periodic in a way no particle account made natural.]
What separates them
[Reserved: the discriminating predictions, stated before any apparatus. Only a wave theory predicts (i) fringes whose spacing scales as \(\lambda D/d\) and vanishes when either slit is blocked, (ii) a bright point at the exact centre of the shadow of a circular obstacle, (iii) a speed of light lower in the denser medium, measured by Foucault and treated in Electromagnetic Waves and Optics, and (iv) a diffraction limit on the resolving power of every instrument. The transversality of the vibration is not predicted by either and is forced by the polarization data of Section 67.4.]
Young: the double slit (1801–1804)
Apparatus and procedure
[Reserved: Young's arrangement — sunlight admitted through a pinhole in a window shutter, a narrow card or a pair of slits splitting the beam, and the fringe pattern received on a screen or observed with a lens [Young:1804]; the coherence requirement met by the common source, which is why the pinhole cannot simply be widened; the slit separation, slit-to-screen distance and fringe spacing as the three lengths that must be measured, all in metres; the later, cleaner realizations that removed the finite source size — Fresnel's biprism and mirrors [Fresnel:1826] and Lloyd's single-mirror geometry [Lloyd:1834], which adds the extra half-wave of reflection.]
Observations
[Reserved: the table to be given — slit separation \(d\), screen distance \(D\) and measured fringe spacing \(\Delta y\) with uncertainties, and the wavelength inferred from \(\lambda=d\,\Delta y/D\); Young's own values, about \(700\,\mathrm{nm}\) at the red end and \(420\,\mathrm{nm}\) at the violet end, to be compared with the modern sodium doublet at \(589.3\,\mathrm{nm}\); the disappearance of the pattern when either slit is covered; and the checks that the spacing is proportional to \(D\) and inversely proportional to \(d\) [Young:1804].]
The two-slit pattern turns light into a measurable length. Three quantities are read with a scale — the separation \(d\) of the apertures, the distance \(D\) to the screen, and the spacing \(\Delta y\) of the fringes — and they return a wavelength. The values so obtained run from about \(700\,\mathrm{nm}\) at the red end of the visible spectrum to about \(420\,\mathrm{nm}\) at the violet end [Young:1804], which is why colour is a physical variable and not merely a sensation. Two checks establish that the reading is what it is claimed to be: the fringe spacing is found proportional to \(D\) and inversely proportional to \(d\), as it must be if it is a path-difference effect, and the pattern vanishes entirely when either aperture is covered [Young:1802] [Young:1804].
Derivation. The theory being used is the two-beam superposition of Electromagnetic Waves and Optics, which gives a fringe spacing \(\lambda D/d\) for \(d\ll D\). What the experiment needs is that relation inverted,
in which every quantity on the right is a length measured with a scale, so the wavelength is obtained without any optical constant, any calibration against another wavelength, or any assumption about the speed of light.
The error budget follows from Equation (67.1) by taking logarithms and differentiating, which turns a product into a sum of relative uncertainties combining in quadrature:
The dominant term is the fringe spacing, which is the smallest of the three lengths, and the standard remedy is in the geometry of the apparatus rather than in the instrument: measure across \(N\) fringes and divide, which divides the relative uncertainty of \(\Delta y\) by \(N\) while leaving the other two terms alone. This is the reason the two-slit determination remains respectable although each individual reading is crude.
∎Interpretation
[Reserved: the general law of interference as Young stated it [Young:1802] — superposition of two disturbances from a common source with a path difference — and the inference that light adds amplitudes, not intensities, so that light plus light can give darkness; why no emission theory accommodates this; the residual objections of the period (the finite width of the fringes, the “inflection” account) and what answered them; the theory tested lives in Electromagnetic Waves and Optics.]
Fresnel and the Poisson–Arago spot (1818–1819)
Apparatus and procedure
[Reserved: the prize memoir's apparatus [Fresnel:1826] — a point source, an opaque circular disc a few millimetres across, and a screen at a distance of order a metre, with fringe positions read on a micrometer; the requirement of a smooth circular edge, which is what makes the null prediction sharp; Fresnel's own half-period zone construction, combining Huygens' wavelets with Young's interference, and the integral form of it given in Electromagnetic Waves and Optics.]
Observations
[Reserved: the bright point at the exact centre of the shadow of the disc, with an on-axis intensity equal to the unobstructed value for an ideal disc; the measured radii of the external and internal fringes of a straight edge and of the disc, compared point by point with Fresnel's integrals; the tabulated quantities are fringe positions in metres against source and screen distances, with the wavelength as the fitted parameter [Fresnel:1826].]
Interpretation
[Reserved: the episode's methodological content, which is why it is in this treatise. Poisson derived the central bright spot from Fresnel's own theory as a reductio; Arago performed the observation and found it; the prediction was therefore made by an opponent and confirmed — the cleanest instance in classical physics of a theory surviving a test designed to kill it [Fresnel:1826]. The quantitative content is that the diffraction integral, not merely the qualitative idea of waves, is correct.]
The Fresnel–Arago laws and transversality
Apparatus and procedure
[Reserved: polarizers by reflection at the Brewster angle [Brewster:1815] and by double refraction in calcite; the two-slit or biprism interference arrangement of Section 67.2.1 with an independently controllable polarization in each arm; the analyser and the photometric comparison method; Malus's original apparatus for the \(\cos^{2}\) law [Malus:1809].]
Observations
[Reserved: the four Fresnel–Arago laws as measured statements [Arago:1819] — two beams polarized at right angles never interfere; two beams polarized in the same plane interfere as unpolarized light does; two beams from perpendicularly polarized components of unpolarized light do not interfere even when brought to a common polarization; and they do if they derive from a single polarized beam. Alongside them, the transmitted intensity \(I=I_{0}\cos^{2}\theta\) [Malus:1809] and the polarizing angle \(\tan\theta_{B}=n\) [Brewster:1815], both with tabulated angles and intensities and their uncertainties.]
Two beams derived from a common source and superposed do not always produce fringes. If they are linearly polarized at right angles to one another, no fringes appear at any path difference whatever: the screen shows only the sum of the two intensities. If they are polarized in the same plane, they interfere exactly as unpolarized light does. Two beams obtained as the perpendicularly polarized components of unpolarized light do not interfere even after being brought back to a common plane of polarization — but two beams obtained the same way from a single polarized beam do [Arago:1819]. These four statements are measured facts, and taken together they force a conclusion about the nature of the disturbance that no amount of interference data alone could give.
Derivation. What a screen records is the time-averaged square of the total disturbance. If the two beams contribute \(\vect{E}_{1}\) and \(\vect{E}_{2}\) at a point,
and all the interference lives in the last term. It is a scalar product, so it vanishes identically — at every path difference, for every amplitude — whenever the two disturbances are orthogonal as vectors. That is the first law, and with it the second: parallel disturbances give the full cross term back.
The conclusion about transversality is now forced. A longitudinal disturbance in a beam has exactly one direction available to it, along the ray, so two beams travelling together could never be orthogonal and the first law could never be observed. The observation therefore requires the disturbance to lie in the plane perpendicular to the propagation direction, where two independent directions exist — and it requires nothing else, which is what makes the inference so strong.
The third and fourth laws add the second ingredient, coherence. The cross term in Equation (67.3) survives averaging only if the phase relation between the two beams is steady. In unpolarized light the two transverse components carry no fixed phase relation, so \(\avg{\vect{E}_{1}\cdot\vect{E}_{2}}\) averages to zero however the polarizations are afterwards rotated into alignment; in a polarized beam they do carry one, and the fringes return.
∎Interpretation
[Reserved: the inference, drawn by Young and by Fresnel, that the vibration is transverse to the propagation direction, since a longitudinal disturbance has no plane to be at right angles to; the amplitude theory of reflection built on it [Fresnel:1832], whose quantitative predictions are the Fresnel coefficients of Electromagnetic Waves and Optics; the cost to the mechanical aether, which must then be a solid, feeding Experiments: Light, the Aether, and Time; and the later mechanical demonstration that a circularly polarized beam carries angular momentum [Beth:1936], which the transverse-wave account of Generalized Classical Field Theory predicts quantitatively.]
Diffraction gratings and wavelength metrology
Fraunhofer's gratings
[Reserved: the wire and ruled gratings with which Fraunhofer turned diffraction into measurement [Fraunhofer:1821] [Fraunhofer:1823]: the grating equation \(d(\sin\theta_{m}-\sin\theta_{i}) =m\lambda\), the first absolute wavelength determinations, and their attachment to the dark lines of the solar spectrum he had catalogued [Fraunhofer:1817], so that a line acquires a number in metres rather than a letter. The apparatus, the angle measurement with a theodolite, and the resulting wavelengths with uncertainties are the content of this section; the spectroscopy they founded is Atomic Models and Spectra.]
A screen ruled with a large number of equally spaced parallel apertures does not merely spread light: it concentrates it into a set of sharp orders at angles that depend on the wavelength, and the orders grow sharper — not merely brighter — as more rulings are illuminated. The angles obey
with \(d\) the ruling interval and \(\theta_{i}\), \(\theta_{m}\) the angles of incidence and diffraction from the normal. Since \(d\) is a mechanical length and \(\theta_{m}\) is read on a divided circle, the wavelength that results is absolute [Fraunhofer:1821]. Applied to the fixed dark lines of the solar spectrum [Fraunhofer:1817], this replaces the letters by which those lines were labelled with numbers in metres.
Derivation. Take a plane wave incident at \(\theta_{i}\) on a row of apertures spaced by \(d\), and look in the direction \(\theta_{m}\). Two adjacent apertures are reached by wavefronts whose paths differ by \(d\sin\theta_{i}\) on the incoming side, and the paths from them to a distant point in the direction \(\theta_{m}\) differ by \(d\sin\theta_{m}\); the net path difference between the contributions of neighbouring apertures is therefore \(d(\sin\theta_{m}-\sin\theta_{i})\), the same for every neighbouring pair. All \(N\) contributions arrive in phase exactly when that difference is a whole number of wavelengths, which is Equation (67.4).
The sharpening with \(N\) is the part that makes the instrument. Away from an order, the \(N\) contributions have phases in arithmetic progression with common difference \(\delta=2\pi(\sin\theta_{m}-\sin\theta_{i}) d/\lambda\), and their sum is a geometric series whose magnitude is \(\abs{\sin(N\delta/2)/\sin(\delta/2)}\). This has principal maxima of height \(N\) where Equation (67.4) holds, and its first zero at \(N\delta/2=\pi\): the angular width of an order therefore falls as \(1/N\) while its height rises as \(N\). That is why a grating with many thousands of rulings separates lines that a prism merely displaces, and it is the origin of the resolving power measured in Section 67.5.3.
∎Rowland's ruling engine and concave grating
[Reserved: the screw-driven ruling engine and the concave grating that made the spectrograph a precision instrument [Rowland:1882]; line densities of order \(600\) per millimetre over apertures of order \(0.1\,\mathrm{m}\) (Rowland's specifications were quoted in lines per inch and are converted here, per the SI rule of the front matter); periodic-error “ghosts” as the signature of an imperfect screw, and the Rowland circle geometry that removes the need for collimating optics; the resulting solar wavelength tables, which were the working standard for decades.]
Resolving power measured
[Reserved: the chromatic resolving power \(\lambda/\Delta\lambda=mN\) of a grating and its verification on known doublets, the sodium pair being the standard test object; the Rayleigh criterion as the convention used to state it [Rayleigh:1879]; the imaging counterpart, Abbe's diffraction limit \(d=\lambda/2n\sin\alpha\) for a microscope [Abbe:1873], with the measured demonstration that no improvement in lens quality passes it; the honest statement that this bound constrains linear imaging only, which is what Section 67.9.3 exploits.]
Resolution is bounded by the wave nature of light and not by the quality of the optics. A grating separates two wavelengths only if they are further apart than a definite fraction of the wavelength, the limit rising in proportion to the order and to the number of illuminated rulings and to nothing else [Rayleigh:1879]; the sodium doublet is the standard test object on which this is checked. In imaging the same bound appears as a smallest resolvable separation, of the order of the wavelength divided by twice the numerical aperture, and it is reached in practice: beyond that point improving a lens changes nothing, which is a measured statement about real microscopes and not a theoretical caution [Abbe:1873].
The diffraction limit: derive the chromatic resolving power of a grating from the angular width of an order — the wavelength divided by the order times the number of rulings — using the \(N\)-aperture sum and the convention that two lines are resolved when the maximum of one falls on the first zero of the other. Then derive the imaging bound from the requirement that at least two diffracted orders from a periodic object be collected by the objective, which gives the smallest resolvable period as the wavelength divided by twice the numerical aperture, and state the hypotheses — linearity in the illumination, far-field collection — on which the bound depends.
The same experiment with X-rays
[Reserved: a crystal as a three-dimensional grating with spacing of order \(0.1\,\mathrm{nm}\), and the Laue photographs that demonstrated both the wave nature of X-rays and the periodicity of crystals in a single exposure [Friedrich:1912]; the geometry, the observed spot patterns and the wavelengths inferred; the crystallography that follows belongs to Phonons and Lattice Dynamics, and the same diffraction argument applied to matter waves is Section 67.9.2.]
Interferometry
The Michelson interferometer
[Reserved: the beam-splitter geometry, the compensating plate, and the stone float on mercury that isolated it from vibration [Michelson:1887]; fringe counting as a length measurement in units of \(\lambda/2\); the coherence-length limitation set by the source, which is why the arms had to be near-equal before lasers; the sensitivity estimate in metres per fringe, which is the number that makes the null result of Experiments: Light, the Aether, and Time meaningful.]
The metre in wavelengths of light
[Reserved: Michelson and Benoit's determination of the international prototype metre in wavelengths of the red cadmium line, \(1\) metre \(=1553163.5\) wavelengths, obtained by a chain of étalons each twice the previous [Michelson:1895]; the error budget of that chain; the consequence that a length standard could in principle be reproduced anywhere, realized in the successive SI definitions and superseded by the fixed value of \(c\) [BIPM:2019]; the modern chain runs through the frequency measurements of Experiment: Precision Spectroscopy and Atomic Clocks.]
The wavelength of light emitted by a specified atomic transition is the same everywhere and at every time, so it can serve as a unit of length that no artefact is needed to embody. The international prototype metre was expressed in such units by counting interference fringes through a chain of étalons, each twice the length of the one before, and the result — \(1\) metre \(=1553163.5\) wavelengths of the red cadmium line — was reproducible to a few parts in \(10^{7}\) [Michelson:1895]. A material standard could therefore in principle be dispensed with, which is what the successive redefinitions of the SI metre eventually did [BIPM:2019].
Derivation. In a two-beam interferometer the two arms are traversed twice, so displacing one mirror through a distance \(x\) changes the optical path difference by \(2x\). The intensity at the output returns to the same value each time that difference increases by one wavelength, so the number \(N\) of fringes counted while the mirror moves is
A length is thereby converted into an integer plus a measured fraction, and the measurement is a count, which is why its precision is not limited by the resolution of any scale.
The chain of étalons is forced by the one quantity Equation (67.5) does not control. Fringes remain visible only while the path difference stays within the coherence length of the source, which even for a good spectral line falls far short of a metre — and a metre contains several million half-wavelengths, which cannot be counted in one pass. Each étalon is therefore compared with one of half its length by counting the fringes over the difference, and only the shortest is compared directly with the wavelength; the uncertainty of the whole chain is the accumulation of those successive comparisons, which is the error budget the experiment must present.
∎Light in a moving medium
[Reserved: Fizeau's interferometer with counter-propagating beams in two tubes of flowing water [Fizeau:1851]; the measured fringe shift, of order a fifth of a fringe for water at a few metres per second over path lengths of order a metre, and the drag coefficient \(1-1/n^{2}\) extracted from it; the reading of the result as a wave-optics measurement here, with its relativistic reinterpretation as the velocity addition law reserved for Experiments: Light, the Aether, and Time.]
Rotation: the Sagnac effect
[Reserved: a closed interferometric loop on a turntable, in which counter-propagating beams acquire a phase difference proportional to the enclosed area and the rotation rate [Sagnac:1913]; the measured fringe shift versus rotation rate; the modern descendants — ring-laser and fibre-optic gyroscopes with resolutions far below the Earth's rotation rate — and the statement that the effect is kinematic and common to every wave, which is why it is not evidence about the aether.]
An interferometer whose two beams travel in opposite senses around a closed loop shows a fringe displacement when the whole apparatus is rotated in its own plane. The shift reverses with the sense of rotation, grows in proportion to the rotation rate and to the area enclosed by the loop, and requires no external reference of any kind — the instrument detects its own rotation while sealed [Sagnac:1913]. In terms of the enclosed area \(A\) and the rotation rate \(\Omega\) about the normal to the loop,
Modern ring-laser and fibre-optic gyroscopes built on this observation resolve rotation rates far below that of the Earth.
Derivation. Take the loop circular of radius \(R\) for simplicity, with the beam splitter fixed on it, and work in the non-rotating frame, in which the light travels at \(c\) in both senses. While a beam goes round, the splitter itself moves: the co-rotating beam must cover the circumference plus the distance the splitter has advanced, and the counter-rotating beam the circumference minus it. Their transit times therefore satisfy \(ct_{\pm}=2\pi R\pm\Omega R\,t_{\pm}\), so
with \(A=\pi R^{2}\) and \(\Omega R\ll c\) in any practicable instrument. Multiplying by the angular frequency \(\omega=2\pi c/\lambda\) converts the time difference into the phase difference Equation (67.6).
Two things are worth reading off the derivation. Nothing in it refers to a medium: the same argument, with \(c\) replaced by the appropriate propagation speed, applies to sound and to matter waves, which is why the effect is kinematic and is not evidence for or against an aether — a point on which the original interpretation was mistaken. And the result depends on the enclosed area rather than on the shape of the loop or the position of the axis, which is what makes a coiled optical fibre of many turns a practical gyroscope: \(N\) turns multiply \(A\), and therefore the signal, by \(N\).
∎Stellar interferometry
Michelson's beam interferometer
[Reserved: the method of measuring an angular diameter from the baseline at which the fringe visibility first vanishes [Michelson:1920], and its execution on the \(2.5\,\mathrm{m}\) Mount Wilson telescope with an outrigger beam of about \(6.1\,\mathrm{m}\) (the original specification was in feet and is converted here); the measured angular diameter of \(\alpha\) Orionis, \(2.3\times 10^{-7}\) radians — about \(47\) milliarcseconds — the first measurement of the size of a star [Michelson:1921]; the visibility-versus-baseline curve as the observable, and the van Cittert–Zernike coherence theory behind it [vanCittert:1934] [Zernike:1938], developed in Electromagnetic Waves and Optics.]
Intensity interferometry
[Reserved: Hanbury Brown and Twiss's measurement of the angular diameter of Sirius by correlating intensity fluctuations at two separated detectors rather than amplitudes [HanburyBrown:1956], which is insensitive to atmospheric phase and so allows long baselines; the measured \(3.3\times 10^{-8}\) radians, about \(6.9\) milliarcseconds; and the controversy the result provoked, since photon correlations are exactly the observable that Quantum Optics and the Photon shows to distinguish classical from non-classical light — the experiment is the origin of quantum optics and is cross-referenced there.]
Gabor: holography and the recording of phase (1948, 1962)
Every measurement in this chapter so far has been a measurement of intensity, because that is all a photographic plate, a photocell or a retina can report: a square-law detector integrates the square of the field, and the phase of the wave that reached it is nowhere in the record. An image is what survives that loss, and it is very much less than the wave was. Gabor's proposal was to stop discarding the phase [Gabor:1948]. Superpose the wave scattered by an object on a mutually coherent reference wave before the detector, and the phase difference between them becomes the position of a fringe — an intensity, which a plate can record. The developed plate then holds the whole wave, amplitude and phase together, whence the name; and re-illuminating it releases that wave again.
The immediate aim was not three-dimensional pictures. Gabor wanted to correct the spherical aberration of the electron microscope, which no electron lens of the period could be made free of, by recording the aberrated electron wave and undoing the aberration afterwards, at leisure and optically, on the reconstruction [Gabor:1949]. That programme proved hard, and the light-optical demonstration built to test the principle carried a defect — a second, unwanted image superposed on the first — which kept the method a curiosity for fourteen years. Both obstacles fell when a coherent source became available [Maiman:1960] and Leith and Upatnieks brought the reference wave in at an angle [Leith:1962] [Leith:1964], and, independently, when Denisyuk recorded the two waves running in opposite senses through a thick emulsion [Denisyuk:1962]. The 1971 Nobel Prize in Physics was awarded to Gabor for the invention and development of the holographic method [Gabor:1972].
Apparatus
Three arrangements, in the order in which they were built.
In-line. A high-pressure mercury arc filtered to a single line and imaged on a pinhole a few micrometres across, so that the illumination is spatially coherent across the plate. The object is a transparency carrying small opaque marks on a clear ground, standing in the diverging beam a short distance in front of a fine-grain photographic plate. In this geometry the part of the beam that passes undiffracted is the reference wave and the light scattered by the marks is the object wave; source, object and plate lie on one axis. That is not a choice but a constraint: the arc is coherent only over a short path difference, so the two interfering paths must be kept nearly equal, and the only way to keep them so is to let them travel together [Gabor:1949]. For reconstruction the developed plate is returned to a similar diverging beam with the object removed, and the emerging wave is examined with a microscope.
Off-axis. A laser beam divided by a beam splitter into two: one illuminates the object, the other is carried to the plate by mirrors and arrives there at an angle to the object wave. Two things follow from having a source coherent over metres rather than micrometres. The reference no longer has to share a path with the object wave, so it can be inclined; and the object may be a solid body at a working distance, illuminated diffusely, instead of a transparency held against the plate [Leith:1962] [Leith:1964]. The demands move to the emulsion and to the bench. The plate must resolve the fringes formed by two beams inclined at \(\theta\), of period \(\lambda/\sin\theta\), which for visible light and an offset of a few tens of degrees is of order a micrometre, some thousand lines per millimetre — far beyond a pictorial emulsion. And nothing may move by an appreciable fraction of a wavelength while the exposure lasts, so the whole arrangement stands on an isolated table.
Counter-propagating. The reference beam enters the emulsion from the front and goes on to illuminate an object placed behind the plate; the light scattered back from the object returns through the emulsion against it. The emulsion is thick — many wavelengths — so that the standing wave the two beams form is recorded as a stack of fringe surfaces through its depth rather than as a pattern on its face [Denisyuk:1962]. Nothing else is needed: one beam, no beam splitter, and no second path to be matched in length.
Procedure
The exposure is made with the object in place and both waves reaching the plate. The plate is developed so that its amplitude transmittance falls as nearly as possible linearly with the exposure it received: linearity of the photographic response is a hypothesis of the method and not an incidental convenience, and it is secured by working on the straight portion of the characteristic curve of the emulsion and checked by repeating the record at several exposure levels.
Reconstruction is the same arrangement with the object taken away and the plate returned to where it stood, illuminated by the reference wave alone. What is observed is the wave emerging on the far side.
Four controls separate the effect from its alternatives.
-
Expose with the reference blocked. The plate then records the intensity of the scattered light alone, develops to a featureless fog and reconstructs nothing: the record is made by the interference and not by the object light.
-
Displace one mirror through a fraction of a wavelength during the exposure. The fringes wash out and the plate again reconstructs nothing, which locates the stored information in the fringe positions.
-
Break the developed plate. Each fragment, illuminated alone, reconstructs the whole scene.
-
Illuminate with a wave conjugate to the reference — the reference reversed. The emerging wave then converges instead of diverging and forms a real image in space, which can be received on a screen with no lens of any kind.
Observations and data
Gabor's reconstructions were legible: a transparency bearing the names of Huygens, Young and Fresnel was recorded and read back from the plate [Gabor:1949]. Superposed on every one of them, out of focus, lay a second image of the same object which no adjustment of the apparatus removed. The field was of the order of a millimetre across and the reconstruction had to be viewed with a microscope.
With the reference wave inclined, that second image is absent from the direction in which the reconstruction is viewed: the waves leaving the plate travel in distinguishable directions, and an eye placed in one of them receives only the reconstruction [Leith:1962]. With a laser and a diffusely illuminated solid scene the reconstruction shows the scene in depth [Leith:1964]. Two observations make that statement operational rather than impressionistic. Moving the eye across the plate changes the perspective continuously — near objects shift against far ones, and their hidden sides come into view — and a camera photographing the reconstruction must be refocused between the near and the far parts of the scene, each setting leaving the rest blurred. Neither is true of a photograph of the same scene.
The plate itself shows nothing. Under a microscope the record of a diffusely illuminated object is a fine irregular structure of fringes bearing no resemblance to the object: the scene is not stored anywhere in it as a picture, which is why a fragment can still deliver the whole of it.
The counter-propagating record reconstructs under ordinary white light, by reflection, and returns the object in the colour of the line that recorded it [Denisyuk:1962]; tilting the plate away from the recording geometry shifts the reconstructed colour towards the blue. A processed emulsion may also contract, which shifts the colour the same way, so the two must be told apart before any colour is quoted.
| Arrangement | Reference wave | Decisive observation |
|---|---|---|
| In-line [Gabor:1948] [Gabor:1949] | the undiffracted part of the illuminating beam, on the same axis | a wavefront is recovered from an intensity record, with the conjugate wave superposed on it |
| Off-axis [Leith:1962] [Leith:1964] | a separate laser beam brought to the plate at an angle | the reconstructed waves separate in direction; a diffusely illuminated solid scene reconstructs with full parallax |
| Counter-propagating [Denisyuk:1962] | the illuminating beam entering a thick emulsion from the opposite side | a volume record selects its own wavelength and reconstructs in white light |
A wave scattered by an object and superposed at a photographic plate on a mutually coherent reference wave leaves a record which, re-illuminated by that reference wave alone, emits a wave agreeing with the original object wave in amplitude and in phase up to a single real constant. An observer looking through the plate sees the object where it stood, in depth and with parallax, although the object is absent and the plate carries no picture of it [Gabor:1948] [Gabor:1949] [Leith:1964]. Three further waves leave the plate at the same time, one of them carrying the complex conjugate of the object wave.
Derivation. Let \(O\) and \(R\) be the complex amplitudes of the object and reference waves in the plane of the plate, both derived from one source and so mutually coherent. What the plate integrates is
of which the first two terms are the separate intensities and the last two are the interference. The phase of \(O\) has not been lost: it stands in Equation (67.8) as the phase of the cross terms relative to \(R\), that is, as the position of the fringes, and a displacement of fringes is a difference of intensity from place to place, which is exactly what a square-law detector does record.
Develop the plate on the linear part of its characteristic, so that its amplitude transmittance is \(t=t_{0}-\beta I\) with \(t_{0}\) and \(\beta\) real constants, and illuminate it with the reference wave alone. The transmitted field is
Four waves therefore leave the plate. The first is the reference wave attenuated; the second is that same wave modulated by the object's own intensity, a halo about it; the third and fourth are the interesting ones.
The third term is the assertion. For a uniform reference \(\abs{R}^{2}\) is a real constant over the plate, so this term is \(O\) itself multiplied by a number: every ray direction, every curvature and every relative phase of the original object wave is present in it unaltered. An eye or a lens placed in that wave receives exactly what it would have received from the object, and therefore sees the object.
The fourth term is the price. It carries \(O^{*}\), the object wave with its phase reversed, which converges where \(O\) diverged and so forms a real image at the mirror position of the object; superposed on the reconstruction it is a second, unwanted image. It cannot be developed away, because Equation (67.8) is a real quantity and its two cross terms are complex conjugates of one another: no record of \(R^{*}O\) that a square-law detector is able to make omits \(R\,O^{*}\).
∎In the in-line arrangement the four waves of Equation (67.9) travel in nearly the same direction, and the conjugate wave is seen superposed on the reconstruction as an out-of-focus ghost that no adjustment removes; it is present in every reconstruction reported from that geometry [Gabor:1949]. When the reference wave is brought in at an angle to the object wave the four leave the plate in different directions and may be looked at one at a time, so that the reconstruction is seen alone [Leith:1962]. The separation appears only above a threshold inclination, and the threshold grows with the angular extent of the object.
Derivation. Put the plate in the plane \(z=0\) and describe the object wave there by its angular spectrum in the plate coordinate \(x\), supported in \(\abs{\nu}\le B\). The bound \(B\) is fixed by the angular extent of the object as seen from the plate, since a point of it lying at an angle \(\varphi\) from the axis contributes the spatial frequency \(\sin\varphi/\lambda\). Let the reference be a plane wave inclined at \(\theta\) in the same plane,
Multiplying a function by \(\ee^{2\pi\ii\alpha x}\) translates its spectrum by \(\alpha\), so the four terms of Equation (67.9) occupy four bands of spatial frequency. The attenuated reference is a single line at \(\nu=\alpha\). The halo carries \(\abs{O}^{2}\), an autocorrelation and hence of width \(2B\), on the same carrier: it fills \(\abs{\nu-\alpha}\le 2B\). The reconstruction carries \(O\) with no carrier at all and fills \(\abs{\nu}\le B\). The conjugate carries \(O^{*}\) on the doubled carrier \(R^{2}\) and fills \(\abs{\nu-2\alpha}\le B\).
The reconstruction is free of the others exactly when its band meets neither of theirs, which requires \(\alpha-2B>B\) from the halo and \(2\alpha-B>\alpha+2B\) from the conjugate. Both reduce to the same condition,
Setting \(\alpha=0\) recovers the in-line case: all four bands are then centred on zero, they overlap whatever the object, and the ghost is unavoidable — which is the observation.
Two consequences are worth reading off Equation (67.11). The threshold is proportional to \(B\) and so to the angular extent of the object, so a larger object demands a more oblique reference; the separation is not free. And the fringe period the emulsion must resolve is \(1/\alpha<1/(3B)\), so the record demands of the emulsion a spatial frequency at least three times the highest the object itself presents. A plate good enough to photograph the object is therefore not good enough to hologram it, by a definite factor — which is why the method needed emulsions that ordinary photography never asked for.
∎A hologram of a diffusely illuminated scene is not an image of it; the developed plate shows only fine irregular fringes. Any fragment of it, illuminated by the reference wave alone, reconstructs the entire scene — at a coarser resolution and over a narrower range of viewing directions, but complete [Leith:1964]. The reconstruction is three-dimensional in the operational sense: moving the eye changes the perspective, and a camera must be refocused between the near and the far parts of the scene.
Derivation. Diffuse illumination is what makes the first statement true, and it is the only ingredient needed for it. A diffusing surface scatters into a wide cone, so light from every point of the object reaches every point of the plate; the plate is therefore not a plane of conjugate image points, as it would be behind a lens, but a surface each point of which carries a contribution from the whole object. Restricting attention to a fragment of width \(a\) removes no object point from the record.
What the fragment does remove is stated by the same wave theory that gave the diffraction limit of Phenomenon 67.4. On reconstruction the fragment is the aperture through which the reconstructed wave passes, so it is the exit pupil of the reconstruction, and the angular resolution with which the scene is seen degrades to about \(\lambda/a\). The range of directions from which the reconstruction can be viewed is the range that fragment subtends at the object, about \(a/z\) for a scene at distance \(z\), so the parallax available shrinks in proportion to the fragment as well. Both are aperture losses; neither is a loss of scene, which is the asymmetry the observation reports.
The depth requires no separate argument. By Phenomenon 67.7 the reconstructed wave is the object wave, so it carries the curvature that the wave from each object point had when it arrived: a point at distance \(z_{1}\) reconstructs as a wave diverging from \(z_{1}\), and a point at \(z_{2}\) as one diverging from \(z_{2}\). An eye or a camera must accommodate separately on the two, and displacing the pupil across the wave changes which rays enter it, which is parallax. Depth in the reconstruction is not a rendering of depth; it is the same optical fact as depth in the scene.
∎When the reference wave and the object wave traverse a thick emulsion in opposite senses, the developed plate reconstructs the object under ordinary white light — a filament lamp, or the sun — and returns it in the colour of the monochromatic light that recorded it; the reconstruction is seen by reflection, on the same side as the source [Denisyuk:1962]. Tilting the plate away from the recording geometry moves the reconstructed colour towards the shorter wavelengths.
Derivation. Two plane waves of vacuum wavelength \(\lambda\) running in opposite senses along \(z\) in a medium of refractive index \(n\) superpose to a standing wave whose intensity varies as \(\cos^{2}(knz)\) with \(k=2\pi/\lambda\). Its antinodes are planes separated by
and in an emulsion many wavelengths deep the development records not one fringe pattern on a surface but a stack of such planes through the volume: a set of weakly reflecting layers of period \(\Lambda\).
Illuminate that stack with white light at normal incidence. Each layer returns a small reflected amplitude, and the reflections from successive layers differ in path by \(2n\Lambda\), so they add in phase only for the wavelength \(\lambda'\) with \(2n\Lambda=\lambda'\) — which by Equation (67.12) is \(\lambda'=\lambda\), the recording wavelength. Every other wavelength present in the illumination is returned by successive layers with phases that spread over a full cycle within the depth of the emulsion, and cancels. The record selects its own recording wavelength out of a broadband source, so neither a filter nor a monochromatic lamp is needed to read it.
The tilt follows from the same condition taken at an angle \(\vartheta\) inside the emulsion, where the path difference between successive layers is \(2n\Lambda\cos\vartheta\) and therefore \(\lambda'=\lambda\cos\vartheta<\lambda\): the reconstruction moves towards the blue, as observed. Any processing that contracts the emulsion reduces \(\Lambda\) and shifts the colour the same way, which is why the two have to be separated before a wavelength is reported.
∎Diffraction efficiency of a hologram: what fraction of the reconstructing beam is delivered into the reconstruction, derived for a thin absorption record and a thin phase record by expanding the transmittance of a sinusoidal grating, and for a thick phase record by the coupled-wave treatment in which two waves exchange energy under the Bragg condition. What has to be established is that only a volume record can send almost all of the incident light into a single order — which is what makes a white-light reflection hologram bright enough to look at — while the thin records are bounded well below that.
Interpretation
The result is about detectors, not about light. Nothing in the four phenomena above goes beyond the superposition principle of Section 67.2 and the diffraction theory of Section 67.5: a hologram is an interference pattern, and a reconstruction is diffraction from it. What the experiment settles is that the phase discarded by every intensity detector is not lost information about the world but recoverable information about the wave, and that a medium with no memory of the object's shape — a uniform emulsion — can be made to hold it. This is why the experiment belongs beside Young and Fresnel rather than among the modern repetitions of Section 67.9: it uses no quantum optics, the reconstruction is a classical field, and the derivations above apply to any wave for which a coherent reference can be arranged.
Three points of method are worth separating out.
First, the second image was not a defect of workmanship. The derivation of Phenomenon 67.7 shows that any real record of the interference contains both cross terms, so the conjugate wave is a consequence of using a square-law detector and could not have been removed by better plates or better alignment. It was removed instead by changing the geometry so that the two terms leave the plate in different directions — a solution in the space of the experiment rather than in the space of its components.
Second, the fourteen-year delay was a coherence limit and not a conceptual one. Equation (67.11) requires the reference to be inclined, and an inclined reference travels a path different from the object wave's; a source coherent only over a short path difference cannot supply one. Here the coherence theory of [vanCittert:1934] [Zernike:1938] appears as a hard constraint on an apparatus: the visibility of the fringes that constitute the record is the mutual coherence of the two beams, and a record of zero visibility is no record at all. A hologram may be read, on this view, as a measurement of the mutual coherence function carried out at every point of the plate at once.
Third, holography does not evade the diffraction limit of Phenomenon 67.4, and it is worth saying why not, since it is often supposed to. The resolution of a reconstruction is set by the aperture of the plate exactly as the resolution of an image is set by the aperture of a lens — that is the fragment result of Phenomenon 67.9. What holography escapes is a different loss, the loss of phase, which is not an aperture effect at all. Gabor's own aim turned on precisely that distinction: an aberration is a phase error, so a recorded wavefront can have its aberration subtracted after the event, which is impossible once the intensity alone has been kept [Gabor:1949]. Carrying that programme out with electron waves proved harder than the light-optical demonstration and came later; the demonstration itself, and the method built on it, are what the prize recognised [Gabor:1972].
Primary references
The proposal [Gabor:1948] and its full development with the in-line experiments [Gabor:1949]; the off-axis reference wave and its reading as a carrier-frequency communication problem [Leith:1962], and the laser reconstruction of diffusely illuminated solid objects [Leith:1964]; the counter-propagating volume record [Denisyuk:1962], published as a note in the Doklady of the USSR Academy of Sciences and available in English translation in Soviet Physics Doklady; the source that made the off-axis geometry practicable [Maiman:1960]; and Gabor's own retrospective account of the twenty-three years [Gabor:1972]. The scalar diffraction theory used throughout is that of [Born:1999], and the coherence theory invoked above is [vanCittert:1934] [Zernike:1938].
Modern repetitions and precision
One photon at a time
[Reserved: Taylor's exposure of a diffraction pattern through smoked-glass attenuators over three months, with at most one photon in the apparatus at a time and no change in the pattern [Taylor:1909]; the modern version with a heralded single-photon source, an anticorrelation measurement at a beam splitter demonstrating particle behaviour and an interferometer demonstrating wave behaviour with the same source [Grangier:1986]; the quantities are the second-order correlation at zero delay and the fringe visibility, with their classical bounds stated explicitly; the theory is Quantum Optics and the Photon.]
The interference pattern does not require many quanta to be present together. Attenuated until at most one photon is in the apparatus at any moment, and exposed for months, a diffraction pattern comes out unchanged [Taylor:1909]; with a heralded single-photon source the same beam that shows particle-like anticorrelation at a beam splitter shows full-visibility fringes in an interferometer [Grangier:1986]. The same holds for matter. Electrons sent one at a time through a biprism arrive as individual localized dots, apparently at random, and the accumulated dots build the fringe pattern [Tonomura:1989]; and molecules of some seven hundred atomic mass units show fringes at the de Broglie wavelength as well [Arndt:1999]. The pattern is therefore a property of each quantum separately and not of the beam, and this is the precise point at which the wave account of this chapter stops being the whole story.
Single-quantum interference: the derivation is quantum and belongs to the later parts. What must be shown here is negative and can be made precise — that no classical field theory reproduces the combination of a single localized detection with a fringe pattern of full visibility, since the second-order correlation at zero delay is bounded below by unity for any classical field, while the measured value for a heralded single photon falls well below it. The wave account of this chapter must therefore be read as the high-occupancy limit of something else.
Matter-wave double slits
[Reserved: Jönsson's electron interference through micro-ruled multiple slits [Joensson:1961]; Tonomura's single-electron buildup, in which the pattern emerges dot by dot from a biprism [Tonomura:1989]; interference of C\(_{60}\) molecules [Arndt:1999], molecules of \(720\) atomic mass units showing the same fringes; the de Broglie wavelength inferred in each case and its comparison with \(h/p\), which is the measurement rather than the illustration; the theory is Matter Waves.]
Beyond the diffraction limit
[Reserved: why Abbe's bound [Abbe:1873] is not violated by stimulated-emission-depletion microscopy [Hell:1994] and its relatives — the imaging is nonlinear in the illumination, so the linear-optics theorem does not apply; near-field microscopy as the other route; the measured resolutions achieved, in nanometres; and the general lesson about the exact hypotheses a “fundamental limit” rests on.]
Interferometers as detectors
[Reserved: the line from Section 67.6.1 to the kilometre-scale, power-recycled, Fabry–Pérot-armed Michelson interferometers of Advanced LIGO [Aasi:2015], whose measured strain sensitivity of order \(10^{-21}\) corresponds to a differential arm-length change of about \(10^{-18}\,\mathrm{m}\) — a fringe fraction of order \(10^{-12}\) — and the first detection made with it [Abbott:2016]; the noise budget (seismic, thermal, shot) as the modern form of Michelson's error analysis; the physics measured is Gravitational-Wave Theory and the experiment proper is Experiment: Gravitational Waves.]