FR32 · Thermal near field

Near-Field Thermal Radiation: Probe Above Hot SiC

Why does a hot surface glow in almost one color when you probe within a micron of it? Because up close the field is dominated by surface waves that cannot escape. Silicon carbide supports such waves at one mid-infrared frequency, where its permittivity is minus one, and heat excites them. Their energy stays within about a wavelength of the surface, so within a micron one line rises far above the broad thermal spectrum; at 100 nm it is about 20,000 times the far-field peak.

Lower a probe toward hot silicon carbide and watch its broad thermal spectrum collapse into one sharp line from surface phonon polaritons.

Probe height $2\,\mu\mathrm{m}$

Strongest frequency $62\times10^{12}\,\mathrm{rad\,s^{-1}}$

Peak over far-field peak $4.24\times$

Energy in bound waves $71.2\%$

Probe the hot surface

Probe height

2\,\mu\mathrm{m}

Sample temperature

300\,\mathrm{K}

Hot material

Drag the probe tip, or press anywhere above the surface.

Advanced: material damping

Damping multiplier

1\times

Scales the phonon damping of SiC or the electron damping of gold. Less damping sharpens the line and lets it grow further before the height cuts it off.

Model, units and sources

$\varepsilon(\omega)=\varepsilon_\infty\,\frac{\omega_L^{2}-\omega^{2}-i\gamma\omega}{\omega_T^{2}-\omega^{2}-i\gamma\omega},\quad \gamma=0.9\times10^{12}\,\mathrm{rad\,s^{-1}}$

$\varepsilon_{\mathrm{Au}}(\omega)=1-\frac{\omega_P^{2}}{\omega(\omega+i\gamma)},\quad \lambda_P=136\,\mathrm{nm},\quad \gamma/\omega_P=0.035/9.0$

Spectra are energy densities per unit angular frequency, in $\mathrm{J\,s\,m^{-3}}$ , emitted by the slab into vacuum at zero temperature. Heights between scan points are interpolated in log scale. The probe does not disturb the field.

Shchegrov, Joulain, Carminati & Greffet (2000), PRL 85, 1548 · Joulain et al. (2005), Surf. Sci. Rep. 57, 59 · Lambrecht & Reynaud (2000), gold parameters

Derive the near-field spectrum step by step

Model assumptions

The hot body is a flat, semi-infinite slab described by a local dielectric function: a single Lorentz oscillator for SiC with the parameters of Shchegrov et al. (2000), or a Drude model for gold. The Lab computes the energy density of the field the slab emits into empty space at zero temperature, electric and magnetic parts together, from the fluctuation–dissipation theorem. The probe is a passive point that does not disturb the field, so this is the quantity a probe would sample, not a detector signal. Below about 10 nm a local continuum model is no longer reliable.

Spectrum at the probe

Two frequencies against height

Which waves carry the energy

More numbers and how they are computed

Strongest frequency, four digits: $0$
Lossless surface-mode limit: $0$
Energy density at the probe, 10 to 400 on the axis: $0$
Far-field energy density, same band: $0$
Black surface, same band: $0$
Exact over close-range limit at 166: $0$

Frequencies are in units of $10^{12}\,\mathrm{rad\,s^{-1}}$ . The far-field part does not depend on height; the bound part is integrated over wavevector with adaptive Gauss–Kronrod quadrature to a relative tolerance of $10^{-4}$ . Model version fr32-thermal-near-field-v1.

Physics tutorial

A hot surface that changes color up close

BackgroundFar from a hot plate you see thermal radiation: a broad spectrum set by temperature and emissivity. Right next to the surface there is also a field that never leaves it, made of evanescent waves whose strength dies off within about one wavelength.

Why it mattersNear-field thermal microscopes, heat transfer across nanoscale gaps and thermophotovoltaic converters all work in this region. Silicon carbide is the textbook case because its surface supports phonon polaritons in the mid-infrared, right where room-temperature radiation is strong.

Start with the essentials

Focus question: Why does a hot surface glow in almost one color when you probe within a micron of it?
One-sentence intuition: Thermal currents in the slab feed every electromagnetic mode, including surface modes that cannot reach the far field. Near the frequency where the permittivity equals minus one, SiC has a huge number of such modes at almost the same frequency. Their energy is confined within a distance comparable to one over their wavevector, so it dominates the spectrum only close to the surface.

Core mathematical model

Emitted energy density

u(z,\omega)=\frac{\Theta(\omega,T)\,\omega^{2}}{2\pi^{2}c^{3}}\left[\int_{0}^{\pi/2}\!\sin\theta\,\frac{e_s+e_p}{2}\,d\theta+\int_{k_0}^{\infty}\!\frac{\beta^{3}\,d\beta}{k_0^{3}\,|\gamma|}\left(\operatorname{Im}r_s+\operatorname{Im}r_p\right)e^{-2|\gamma| z}\right]

The first term is ordinary emission into travelling waves, weighted by the directional emissivities. The second counts evanescent waves with $\beta>k_0$ ; they decay as $e^{-2|\gamma|z}$ with $|\gamma|=\sqrt{\beta^{2}-k_0^{2}}$ . It is normalised so that a black surface gives half the cavity value of Joulain et al. (2005), Eq. 47, and the close-range limit is their Eq. 48.

Surface-mode frequency

\operatorname{Re}\varepsilon(\omega_{\mathrm{sp}})=-1\ \Rightarrow\ \omega_{\mathrm{sp}}=\sqrt{\frac{\varepsilon_\infty\omega_L^{2}+\omega_T^{2}}{\varepsilon_\infty+1}}=178.7\times10^{12}\,\mathrm{rad\,s^{-1}}

With $\varepsilon_\infty=6.7$ , $\omega_L=182.7\times10^{12}\,\mathrm{rad\,s^{-1}}$ and $\omega_T=149.5\times10^{12}\,\mathrm{rad\,s^{-1}}$ , this is the line Shchegrov et al. (2000) report for SiC.

Very close to the surface

u(z,\omega)\approx\frac{\Theta(\omega,T)\,\omega^{2}}{4\pi^{2}c^{3}}\,\frac{\operatorname{Im}\varepsilon}{|\varepsilon+1|^{2}}\,\frac{1}{(k_0 z)^{3}}

For heights much smaller than the wavelength the energy density grows as one over the height cubed, and the factor one over the squared distance of the permittivity from minus one picks out the surface-mode frequency. The dashed lines in the height chart are this limit.

Common difficulties

The near field is just brighter blackbody light

Typical misconceptionClose to the surface you see the same spectrum, only stronger.

Better mental modelThe far-field spectrum of SiC even has a dip where its reflectivity is high. Up close the strongest feature sits inside that dip, at a frequency the far field barely contains.

Bound waves carry heat away

Typical misconceptionThe extra energy near the surface is flowing out into space.

Better mental modelEvanescent waves above a single body carry no net energy to infinity; they store energy next to the surface. They carry heat only when a second body is close enough to absorb them, which is the next Lab.

Run the experiment

01
Start far away
Choose Far-field spectrum and look at the spectrum chart.
What to observe: The spectrum is broad, with a dip between about 150 and 180 on the frequency axis where SiC reflects strongly.
02
Predict, then approach
Guess at what height a single line takes over, then run Approach scan or drag the probe down.
What to observe: A line at 178.7 appears near 2 µm and becomes the strongest feature near 1 µm. At 100 nm it is about twenty thousand times the far-field peak.
03
Explain with gold
Choose Gold: no line and compare the mode map with SiC.
What to observe: Gold has no surface mode in the infrared, so its near field grows without a line and is strongest at the lowest frequencies shown.

Near-Field Thermal Radiation: Probe Above Hot SiC

A hot surface that changes color up close

Core mathematical model

Emitted energy density

Surface-mode frequency

Very close to the surface

Common difficulties

The near field is just brighter blackbody light

Bound waves carry heat away

Run the experiment

Start far away

Predict, then approach

Explain with gold

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