Illustrated Physics
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Series

Coherence — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two sources 4 wavelengths apart. Circular wavefronts from two sources, with the lines along which they arrive in step drawn through the pattern. Those lines are where the path difference is a whole number of wavelengths.

    Why two lamps never interfere

    Adding amplitudes is unconditional; fringes are not. What decides is whether the phase difference holds still for longer than a detector takes to record it — and a 10 nm slice of white light holds it for 100 femtoseconds, across a path of 30 micrometres.

    part 1 · optics
  2. How far apart the two paths can be. Fringe visibility against the difference between the two path lengths, for light at 550 nm with a bandwidth of 100 nm. Each curve is the modulus of the Fourier transform of its own line shape, summed over the spectrum here rather than taken from a standard result, and the three shapes have the same width at half height. The conventional coherence length λ²/Δλ is 3.02 µm for this light, and what the curves show is that the convention is a rounding of three genuinely different behaviours: a flat band halves at 1.83 µm, a Gaussian line halves at 1.33 µm, a Lorentzian line halves at 0.68 µm. The flat band comes back — a rectangle's transform rings — and the Lorentzian's tails keep a little visibility very much further out than its width suggests. Nothing here is about the apparatus: the fade is the source forgetting its own phase.

    How far a wave can remember

    Split a beam, delay one half, and put them back together. The fringes are bright while the delay is short and fade as it grows, and the distance at which they die is fixed by nothing but the width of the source's spectral line. Watching them fade is reading the line shape.

    part 2 · optics
  3. 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.

    part 3 · optics
  4. A grain that is not on the object. A speckle pattern, computed as the far field of a circular aperture 421 samples in area filled with random phases — which is what a rough surface does to coherent light, and nothing else. The texture is not a picture of the surface: change the phases and the grains move, but their size and their statistics do not. Five shades are drawn here, from the darkest fifth of the range to the brightest. The measured contrast — the standard deviation of the intensity divided by its mean — is 1.0101, against exactly one for a fully developed speckle, and that is a strong statement: it says the most likely intensity anywhere in this pattern is zero, and that the bright grains are as far above the mean as the dark ones are below. Anybody who has pointed a laser at a wall has seen this and most take it for a property of the wall. It is a property of the light and of the aperture looking at it — including, when the aperture is an eye, of the pupil, which is why the pattern swims when the head moves and why its grain size tells an optometrist about the eye rather than about the wall.

    The grain that is in the light

    Point a laser at a wall and the wall appears to be covered in a fine boiling texture. Nothing on the wall is that size and nothing about the wall decides it: the grain belongs to the aperture looking at it, the statistics are the same for every rough surface there is, and the most likely brightness anywhere in the pattern is zero.

    part 4 · optics
  5. 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.

    part 5 · optics
  6. 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.

    part 6 · optics
  7. A strong oscillator lifts a picowatt to the quantum limit. The signal-to-noise ratio for detecting a 1 pW signal at 1.55 µm in a 1 MHz bandwidth, with a detector of 80 per cent quantum efficiency whose amplifier adds 2 pA/√Hz of noise, against the power of the local oscillator. Detected directly, the signal is 66 dB below the amplifier's noise. Heterodyned, it rises with the oscillator's power — −3.3 dB with a microwatt, 7.4 dB with a hundred — until the oscillator's own shot noise swamps the amplifier's, above about 12 µW, and then it stops rising at 8.0 dB: ηPₛ/(hνB), the number of detected signal photons per resolution time, which no receiver can beat. A stronger oscillator raises the signal and its own noise together.

    The faint light a bright one makes measurable

    A photodetector does not respond to a light wave's field; it responds to the square of it. Shine a strong beam onto the detector together with a weak one at a slightly different frequency, and the square contains a term that is the weak field multiplied by the strong one — oscillating at the difference frequency, carrying the weak light's phase, and as large as the strong beam cares to make it. That product lifts a picowatt out of an amplifier's noise to the limit set by counting photons, turns an optical frequency shift into a radio one, and carries with it a quantum price that decides why radio telescopes and optical telescopes are built so differently.

    part 7 · optics
  8. A scan into a layered sample, with a broad source and a narrower one. The envelope of the interference signal against depth in a sample with five reflecting boundaries (marked), at 60, 72, 150, 158 and 260 µm, scanned with a 50 nm source (solid) and a 10 nm source (dashed), both centred at 840 nm. The broad source, with a 6.2 µm resolution, separates every boundary, including the two that are only 8 µm apart. The narrow one, with 31 µm, merges the pairs into single blurred echoes. Every boundary in a real eye or skin sample is a small change of refractive index that reflects a tiny fraction of the light; the scan is a map of where those changes are.

    The depth a broad spectrum can see

    Interference is usually spoiled by a broad spectrum: many colours drift out of step, and fringes wash out a few wavelengths from where two paths are equal. Turned round, that failure is a ruler. Light whose colours stay in step only when two paths match to within a few micrometres can report the depth of every reflecting layer inside something translucent — a retina, a skin, a coat of paint — to that precision, without cutting it open. Optical coherence tomography reads depth off the width of a spectrum, and it has become one of the most used imaging methods in medicine; its resolution is set by how many colours the light contains, and the finest instruments use sources whose spectra span hundreds of nanometres.

    part 8 · optics
  9. How narrow a laser's line can be, against its power. The Schawlow–Townes limit on linewidth, πhνΓ²n/P with Γ the cavity linewidth and n the inversion factor, against output power on logarithmic axes, for a helium–neon laser with a 30 cm cavity of finesse about 300 (Γ = 1.6 MHz) and for a semiconductor laser a third of a millimetre long (Γ about 30 GHz, n = 2), the latter also with Henry's linewidth enhancement 1 + α² for α = 4 (dashed). At a milliwatt the helium–neon laser's limit is 0.0025 Hz — millihertz, a coherence length of tens of millions of kilometres — and the semiconductor laser's 7.2·10⁵ Hz, 1.2·10⁷ Hz with the enhancement. Both fall in proportion to one over the power: more photons in the mode make each spontaneous kick a smaller fraction of the field. The square of the cavity linewidth is why a short, lossy diode cavity has a line ten billion times broader.

    The width a perfect laser still has

    A laser's light is the nearest thing to a single pure frequency that anything makes, and it is still not one. Every photon spontaneously emitted into the lasing mode adds a kick of random phase, and because nothing in a laser cares what its phase is, nothing pulls the phase back: it wanders, and a wandering phase is a line of finite width. Schawlow and Townes worked out how wide in 1958, before any laser had been built. For a milliwatt helium–neon laser the answer is a few thousandths of a hertz, a coherence length a quarter of the way to the Sun — and no real laser of that kind has ever come close, because mirrors move. For a semiconductor laser the answer is megahertz, and the measured lines were wider still, by a factor nobody expected.

    part 9 · optics

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