FR32 · An interactive course
Fluctuational electrodynamics: heat and force across a narrow gap
Hold two warm plates tens of micrometres apart and they exchange ordinary thermal radiation, never more than two blackbodies would. Bring them to within ten nanometres and the flow is more than a thousand times what two blackbodies could exchange; the same fluctuating fields also pull the plates together hard enough to tilt a micromachined plate. This course derives both from one picture: random currents inside a warm material and the waves they launch.
From a warm surface to a force across the gap
The first chapter writes Planck’s spectrum as the energy of one mode times the number of modes, and links emission to absorption through Kirchhoff’s law. The second shows how fields with a large in-plane wavevector stay bound to a surface and finds the surface wave of silicon carbide. The third uses the fluctuation–dissipation theorem to compute the energy density above a hot surface and the narrow line that appears 100 nm away. The fourth counts the channels that carry heat between two plates and recovers the Stefan–Boltzmann law from black plates. The fifth follows the tunnelling channels down to a 10 nm gap and shows why the flux grows as the inverse square of the gap. The sixth keeps the same fluctuating fields at equal temperatures and turns them into the Casimir force on a torsional oscillator. The last chapter reads the 2000 letter by Shchegrov and co-workers figure by figure.
You need Maxwell’s equations in a uniform medium, complex numbers, the Planck spectrum from an introductory modern-physics course and reading a log–log plot. The fluctuation–dissipation theorem, the Fresnel coefficients and the proximity-force estimate are introduced when they are first needed. FR32
01 · L1
Why hot surfaces glow, and why a good mirror glows weakly
Build Planck’s spectrum from mode energy and mode count, derive the Stefan–Boltzmann law, and use Kirchhoff’s law to explain the emission dip of SiC.
02 · L2
Evanescent waves and the surface phonon polariton of silicon carbide
Fields with too large an in-plane wavevector cling to a surface. Derive the SiC surface-wave dispersion, its decay length and where the modes pile up.
03 · L3
Why a broad thermal spectrum shrinks to one line 100 nm from a hot surface
Use the fluctuation–dissipation theorem for the energy above a hot surface, derive its inverse-cube growth and find the nearly single-colour SiC line at 100 nm.
04 · L3
Radiative heat between two plates, counted channel by channel
Radiative heat between two plates is a sum over channels that each carry at most one share. Recover the Stefan–Boltzmann law and check detailed balance.
05 · L3
Why near-field heat transfer grows as the inverse square of the gap
Count the tunnelling channels between two SiC plates, show why the heat transfer coefficient grows as one over the gap squared, and test damping and materials.
06 · L3
How the Casimir force gives a micromachined plate two swings
Correct the ideal-mirror Casimir pressure for real gold, then find the frequency shift, nonlinearity and hysteresis of the Chan et al. torsional oscillator.
07 · L4
Reading Shchegrov et al. (2000) on why silicon carbide looks different near and far
Read the four-page letter in order, find its pivot in Eq. (7), recompute the line position, the dip and the inverse-cube law, and see what Fig. 1 leaves out.
Experiments along the way
All three experiments evaluate the same formulas the chapters derive; none of them fits or redraws published curves. The probe experiment computes the energy density above a SiC or gold half-space and draws the far-field spectrum and a black surface beside it for comparison. The heat-transfer experiment sums the channels between two plates and draws two black plates as an exact reference. The Casimir experiment combines the proximity-force estimate with the Lifshitz reduction factors for three models of gold and mirrors; the oscillator constants come from the paper by Chan et al., and its measured points are compared in the benchmark record rather than drawn in the Lab.
Sources and what we recomputed
- A. V. Shchegrov, K. Joulain, R. Carminati and J.-J. Greffet, Near-field spectral effects due to electromagnetic surface excitations, Physical Review Letters 85, 1548–1551 (2000), HAL hal-01624761v1
Full text of the published letter in the HAL archive. Eq. (1): the Planck spectrum as mode energy times mode density. Eqs. (2) to (5): thermal currents from the fluctuation–dissipation theorem and the energy density of the emitted electric field. Eq. (6): the surface-wave dispersion relation. Eqs. (7) and (8): the near-field spectrum as mode energy times a local mode density that grows as one over height cubed. Parameters of SiC on p. 1549. Fig. 1: spectra at 1000, 2 and 0.1 µm above SiC at 300 K, with the line at 178.7 in units of 10 to the 12 rad/s. Fig. 3: decay with height at 166 and 210 in the same units. We recomputed the line position, the height law and the decay at 166; we did not digitise the figures.
- K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati and J.-J. Greffet, Surface electromagnetic waves thermally excited: radiative heat transfer, coherence properties and Casimir forces revisited in the near field, arXiv:physics/0504068v2; Surface Science Reports 57, 59–112 (2005)
Version 2, 24 April 2005. Sec. 2: surface waves, the Drude permittivity (Eq. 21) and the SiC oscillator (Eq. 22). Sec. 3.2: the fluctuation–dissipation theorem and the mean oscillator energy (Eqs. 25 to 30). Secs. 4.1 and 4.2: the emitted energy density (Eqs. 44 to 48), emissivity as the half-sum of the two transmission factors and its link to Kirchhoff’s law, and Fig. 6 at 100 µm, 1 µm and 100 nm above SiC. Sec. 7: heat flux between two half-spaces (Eqs. 89 to 93), the growth as the inverse square of the gap, and Figs. 17 and 18, which they reproduce from another paper. Sec. 8: the Casimir force from the same fluctuating fields. The evanescent term of Eq. 45 as printed carries an extra factor of two; we fixed the prefactor from Eq. 48. The prefactor of Eq. 93 is garbled in the text version we read, so we quote only its inverse-square dependence on the gap.
- H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop and F. Capasso, Nonlinear micromechanical Casimir oscillator, arXiv:quant-ph/0109046v2; Physical Review Letters 87, 211801 (2001)
Version 2, 21 September 2001. The device: a 500 µm polysilicon plate, gold-coated, on two torsional rods, with a gold-coated sphere of radius 100 µm. Eq. (1): the equation of motion with the Casimir force expanded to third order. Eqs. (2) and (3): the shifted resonance and the nonlinear coefficients. Eq. (4): the perfect-mirror force gradient used for their fit, with the closest distance 85.9 nm and lever arm 131.0 µm from the fit. Eq. (5): the amplitude equation. Fig. 4 caption: Q of 7150, the four distances, the four values of the nonlinear coefficient, a 39 nm swing and a fixed drive at 2748 Hz in panel (b). Text: in the bistable region the two swings differ by up to a factor of 6. We recomputed the coefficients and the hysteresis; we did not digitise the data.
- A. Lambrecht and S. Reynaud, Casimir force between metallic mirrors, arXiv:quant-ph/9907105v2; European Physical Journal D 8, 309–318 (2000)
Version 2. Eqs. (1) and (2): the Casimir energy and force between perfect mirrors. Eqs. (3) to (6): the plasma wavelength and the reduction factors for force and energy. Eq. (7): the thermal length, a few µm at room temperature. Eq. (8): the proximity-force estimate for a sphere and a plane. Eqs. (14) and (15): the short-distance slopes. Sec. V and table (20): gold with a plasma frequency of 9.0 eV (plasma wavelength 136 nm) and damping 35 meV, and reduction factors at 0.1, 0.5 and 3 µm from tabulated optical data. Text: relaxation changes the force by less than 2 %. We recomputed the asymptotes and the table with model permittivities.
- NIST Reference on Constants, Units, and Uncertainty: speed of light in vacuum, reduced Planck constant, Boltzmann constant and Stefan–Boltzmann constant
The value pages for the four constants. Since 2019 the speed of light, the Planck constant and the Boltzmann constant are exact in the SI, so the Stefan–Boltzmann constant computed from them agrees with the NIST value 5.670374419 to all ten digits listed.