Illustrated Physics
physics, drawn
Concept

Visibility — where it appears

The contrast of an interference pattern, running from one for perfect fringes to zero for none. For a distant source it equals the modulus of the degree of coherence, so measuring where it first vanishes measures the source's angular size.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Fringe contrast against baseline, for four stellar diameters. The visibility of the fringes an interferometer would obtain at 575 nm, against the separation of its two apertures, for uniform discs of angular diameter 10, 20, 47, 100 milliarcseconds. Each curve is 2J₁(πθB/λ)/(πθB/λ), the transform of a uniform disc, and each first reaches zero at 14.47 m for 10 mas, 7.23 m for 20 mas, 3.08 m for 47 mas, 1.45 m for 100 mas. Dividing each of those by λ/θ returns the same number, 1.2197, which is the 1.22 in every textbook and is the first zero of J₁ divided by π — recovered here from the four curves rather than written into them. The practical content is that a smaller star needs a longer baseline, in exact inverse proportion, and that the measurement is of a contrast rather than of a picture. Michelson and Pease found the fringes from Betelgeuse vanishing at a 3.07 m separation in 1920 at a wavelength of 575 nm, which by the same arithmetic is a disc 47.1 milliarcseconds across — and no telescope resolved that star for another seventy years.

The fringe that measures a star

Set two apertures 3.07 metres apart in 1920 and the fringes from Betelgeuse vanish. That single fact gives the star's angular diameter to two significant figures, without ever forming an image of it — because the contrast of a fringe pattern is a Fourier component of the source's own shape.

optics · Coherence The same source measured by amplitude and by intensity. The degree of coherence of a 47 milliarcsecond disc at 550 nm, and its square, against the separation of two apertures. A Michelson interferometer measures the upper curve, because fringe contrast is |γ|. Correlating the intensities at the two apertures instead measures the lower one, because the excess correlation of two thermal beams is |γ|² — the same information about the source, since one curve determines the other, and reaching zero at the same baseline of 2.94 m. What is lost is the phase of γ, which the intensity correlation never sees; what is bought is that a path error of many wavelengths does not matter, because the quantity being correlated is a slow fluctuation of brightness rather than a wave. The half-coherence baselines differ — 1.71 m against 1.25 m — which is the practical statement that the squared curve is the steeper one to measure against.

The correlation that survives what the phase does not

Two telescopes can measure a star's diameter by interfering the light, which requires holding two paths equal to a fraction of a wavelength through an atmosphere that will not hold still. Or they can throw the phase away entirely and correlate the brightness fluctuations, which needs the paths equal to a few metres and works.

optics · Coherence One measurement that separates unpolarised from polarised. What a rotating linear polariser passes, against its angle, for beams of the same total intensity and degrees of polarisation 0, 0.35, 0.7, 1. Every curve has the same average — a polariser passes half of any beam over a whole turn, whatever its state — and they differ only in how deeply they modulate. The depth of the modulation is the degree of polarisation, exactly: a fully polarised beam goes to zero at one angle and a beam with no preferred direction gives a flat line at a half. That is the whole measurement, and it is why "unpolarised" is a statement about a modulation depth rather than about what a wave is doing.

The light with no direction of shaking

Unpolarised light is not a state of a wave; it is the absence of one, and no description of a single wave can represent it. What can is a set of four numbers, all of them powers a detector reads — and they describe every beam there is, including the ones that are neither polarised nor not.

optics · Polarisation The spectrum, and what the interferometer records instead. On the left, a source spectrum: 1 line near 2000 reciprocal centimetres. On the right, what a detector behind a two-beam interferometer reads as the path difference is scanned — the interferogram. It is the cosine transform of the spectrum, so the two panels carry exactly the same information and neither is more fundamental. The fast oscillation is the mean wavenumber; the envelope that decays over about 0.133 centimetres is the reciprocal of the linewidth, which is the coherence length; and where two lines are present, the beat between them is the splitting. Nothing disperses anything anywhere in the instrument.

The fringe and the spectrum are one measurement

An interferometer with no prism and no grating in it measures a spectrum, because what it records as the path difference is scanned is the Fourier transform of the source's spectrum. Coherence length and linewidth are the same fact stated twice, and the resolution is bought in centimetres of travel.

optics · Coherence Minima that are not zeros. The amplitude along a line carrying a wave towards a load and its reflection back, for reflection magnitudes of 0, 0.35, 0.7, 1. With everything reflected the pattern touches zero and is a standing wave in the strict sense. With less than everything it does not: the minima sit at one minus the reflection and the maxima at one plus it, so the pattern is a partial standing wave sitting on a travelling one. The spacing is half a wavelength in every case, and the depth is the only thing that changes — which is why one number, the ratio of the maximum to the minimum, is enough to report the whole pattern.

The node that is not standing still

A wave meeting a perfect reflector makes a standing wave with real nodes. A partial reflection makes something that looks the same and is not: the minima are not zeros, energy flows steadily through them, and the depth of the pattern is a measurement of the load that caused it.

waves · Standing waves

Named alongside it

The objects these essays reach for when they reach for this one.

CoherenceAngular diameterFourier transformInterferenceSuperpositionAntibunchingAperture synthesisBandwidthBaselineBoundary conditionsCircular polarisationCoherence length

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