L4 · Chapter 7

Reading Shchegrov et al. (2000) on why silicon carbide looks different near and far

Sandbox Physics · 2026-10-30

This letter is only four pages long, yet it set the basic picture for later work on near-field thermal radiation: the source stays the same, the closer a probe gets the more modes it collects, and the less the spectrum looks like a black body. Reading it does not require computing Green tensors, but it does require knowing what each equation does in the argument, which numbers you can recompute, and which conclusions rest on figures nobody has digitised.

What the letter asks

In the 1980s Wolf predicted that a spectrum can change as light propagates, but such changes are usually small shifts of a line. Shchegrov and co-workers set out to show a much larger change: the same thermal source can give completely different spectra near and far. The cause is not the statistics of the source but evanescent waves, which exist near the surface and never reach the far zone.

The opening paragraph writes the Planck spectrum as mode energy times mode density. It reads like a textbook reminder, but it sets up everything after it: if the mode density changes with position, so does the spectrum.

Source: A. V. Shchegrov

A reading order

On a first pass, read Eq. (1) and the text around it, then jump to the SiC parameters on p. 1549 and to Fig. 1. Its three panels, at 1000 µm, 2 µm and 0.1 µm, are the phenomenon the letter sets out to explain.

Then read Eq. (7): $I(z,\omega)=\Theta\,N(z,\omega)$ . The authors call it the pivotal point of the paper. It traces the changes in Fig. 1 to a local density of modes that depends on height, in exactly the form of Eq. (1).

Next read Eq. (6) and Fig. 2 to see where the surface-wave dispersion flattens, then Eq. (8) to see how the mode density grows as $z^{-3}$ at short range and resonates where $|1+\varepsilon|$ is smallest. Finish with Fig. 3: at 166 there is a surface wave and an exponential decay between 1 and 5 µm; at 210 there is not.

Eqs. (2) to (5) compute the energy density from the fluctuation–dissipation theorem and the Green tensor. On a first pass, keep only the conclusion: only the integral over the magnitude of the in-plane wavevector needs numerical work. Chapter 3 gives a formula of the same structure.

I(z,\omega)=\Theta(\omega,T)\,N(z,\omega),\qquad N(z,\omega)\;\xrightarrow{\;z\ll\lambda\;}\;\frac{\varepsilon''(\omega)}{|1+\varepsilon(\omega)|^{2}}\,\frac{1}{16\pi^{2}\omega z^{3}}

Eqs. (7) and (8). Here

\varepsilon''

is the imaginary part of the permittivity.

Source: A. V. Shchegrov

Check four numbers yourself

Line position. With the parameters on p. 1549, setting the lossless permittivity to $-1$ gives $178.74\times10^{12}\,\mathrm{rad\,s^{-1}}$ . The letter writes 178.7, and the course model’s peak at 100 nm also falls at 178.7.

Far-field dip. The letter says the far-field spectrum has a wide dip between 150 and 180, where the sample hardly emits. The angle-weighted emissivity is 0.115 at 150, reaches its lowest value of 0.030 at 160, is back to 0.092 at 180 and is 0.19 at 182.7.

Decay at 166. The worked example of chapter 2 finds an energy decay length of 2.30 µm from the dispersion relation; an exponential fit to the full model between 2 and 5 µm gives 2.286 µm.

Height law. Eq. (8) requires the energy density at the line to scale as $z^{-3}$ . In the course model it grows 992 times from 100 nm to 10 nm, within 1% of 1000.

All four numbers use only parameters and formulas printed in the letter. We did not digitise the curves in the figures, so the discussion of the figures below is limited to qualitative features such as where a peak sits and which peak is taller.

Source: A. V. Shchegrov

Fig. 1 plots electric energy

Eq. (5) gives the energy density of the emitted electric field, without the magnetic field. In the far field the two share the energy equally; at short range the magnetic field of p-polarised evanescent waves is weak and nearly all the energy is electric. So at 0.1 µm the figure differs little from the total electromagnetic energy, but at 2 µm it does.

The course model computes both at 2 µm. With the electric field alone, the highest peak of the whole spectrum is at $71.5\times10^{12}\,\mathrm{rad\,s^{-1}}$ and the local peak inside the reststrahlen band is at 176.75, 0.89 of the highest. With the total electromagnetic energy, the highest peak is at 62 and the band peak at 176, only 0.44 of the highest. The difference is magnetic energy, concentrated at low frequency.

Describing 2 µm, the letter says a peak emerges at 178.7. Both calculations do show a local peak appearing in the band, but it is not yet the tallest, and it sits about 1% below 178.7. When comparing the course Lab with this figure, remember that the Lab shows total electromagnetic energy.

There is also a difference in normalisation. Multiply Eq. (8) by $\Theta$ and compare it with Eq. (48) of Joulain et al. (2005): the dependence on $z$ and $\varepsilon$ is identical, but the prefactors differ by a factor of 4. Since nearly all the energy is electric at short range, this is not a physical difference but a difference in how the two papers define a spectral density; we did not trace which factors account for it. The course compares only ratios and peak positions, and the factor cancels in both.

Source: A. V. Shchegrov · K. Joulain

Worked example: how much brighter the line is than 166 at 0.1 µm

Problem. Using only Eqs. (7) and (8), estimate how many times larger the spectral density is at 178.7 than at 166, at 300 K and a height of 0.1 µm. Then compare with the full model.

Model. In Eq. (8) the height is the same for both, so the ratio keeps three factors: $\varepsilon''/|1+\varepsilon|^2$ , $1/\omega$ and $\Theta$ .

Numbers. Chapter 2 found $\varepsilon''/|1+\varepsilon|^2$ to be 7.697 at 178.7 and 0.009646 at 166, a ratio of 798. The frequency factor is $166/178.7=0.929$ . The ratio of $\Theta$ is 0.776: photons at 178.7 carry more energy and are less populated at 300 K. The product is 575.

Check. The full model gives 509 at 0.1 µm, 11% below Eq. (8), and 575 at 10 nm, in agreement with it. The reason is in chapter 3: at 0.1 µm a fair share of the energy at 166 lies outside the quasi-static limit, and Eq. (8) underestimates it.

The example shows a habit worth keeping with letters like this: whether an asymptotic formula can be used depends on which frequency and how close, and the best test is a ratio you can compute directly.

\frac{I(z,178.7)}{I(z,166)}=\underbrace{\frac{7.697}{0.009646}}_{798}\times\underbrace{\frac{166}{178.7}}_{0.929}\times\underbrace{\frac{\Theta(178.7)}{\Theta(166)}}_{0.776}=575

Frequencies in units of

10^{12}\,\mathrm{rad\,s^{-1}}

, temperature 300 K.

Source: A. V. Shchegrov

What the letter does not do

This is a theory letter with no measurement. It ends by saying that such spectral changes should be observable in many solids. A real measurement needs a probe that scatters the near field into the far zone, and the probe itself changes the field it measures; the probe in the letter is an ideal observation point that disturbs nothing.

The model assumes a half-infinite sample with a flat surface and a local permittivity that does not depend on temperature. Joulain and co-workers kept the same assumptions in their later review and extended the method to heat flow and the Casimir force between two bodies, the subject of chapters 4 to 6 of this course.

Source: A. V. Shchegrov · K. Joulain

Try it in the Lab

Redraw Fig. 1 in the Lab

Predict first: at heights of 1000 µm, 2 µm and 0.1 µm, where will the strongest frequency be?
Choose the Far-field spectrum, A line emerges and Almost one color presets in turn (1 mm, 2 µm and 100 nm) and read the strongest frequency and the peak over the far-field peak each time.
Compare with the description of Fig. 1 in the letter. If at 2 µm the tallest peak is not near 178.7, explain it with the section on electric energy.

Redraw Fig. 3 in the Lab

Predict first: of the curves at 166 and 210, which is a straight line between 1 and 5 µm? Which rises faster below 1 µm?
In any preset, look at the Two frequencies against height chart.
Explain the straight stretch at 166 with the dispersion relation of chapter 2, and use Eq. (8) to explain why both curves rise as $z^{-3}$ below 1 µm.

Exercises with explained answers

By Eq. (8), by what factor does the spectral density at the line grow when the probe goes from 0.1 µm to 0.05 µm? Will the full model give more or less?
Show the answer
Eq. (8) gives $2^3=8$ . The full model is about 0.8% above Eq. (8) at 0.1 µm and approaches it as the probe gets closer, so the growth from 0.1 µm to 0.05 µm is slightly less than 8.
The letter says that between 150 and 180 the sample effectively acts as a non-radiating source. Explain this with Kirchhoff’s law, and say why it fails in the near field.
Show the answer
In this band $\operatorname{Re}\varepsilon<0$ , the surface reflects almost everything, and the angle-weighted emissivity falls to 0.030. Emissivity equals absorptivity, so travelling waves carry almost no energy out. But the thermal currents inside keep fluctuating, and the evanescent and surface waves they excite are not bound by this limit; they just cannot reach the far zone. Once a probe comes close it collects these modes, and the same band becomes the brightest part of the spectrum.
Imagine a weakly absorbing material with no surface wave and a constant permittivity of $4+0.4i$ . What is its normal emissivity in the far field? What is its near-field factor $\varepsilon''/|1+\varepsilon|^2$ ? Which sentence of the letter does this illustrate?
Show the answer
The normal reflectance is $|(1-\sqrt\varepsilon)/(1+\sqrt\varepsilon)|^2=0.112$ and the emissivity 0.888: a half-infinite weak absorber is nearly black in the far field. The near-field factor is $0.4/(5^2+0.4^2)=0.0159$ , small. The far-field spectrum follows the emissivity and the near-field spectrum follows this factor, so they weight frequencies differently. This is what the letter says: even without resonant surface waves the near-field spectrum differs from the far-field one, only less dramatically.
The last paragraph of the letter says the mode density of silver resonates at $5.57\times10^{15}\,\mathrm{rad\,s^{-1}}$ , yet no sharp peak is seen unless the sample is above 4000 K. Explain with $\Theta$ from chapter 1.
Show the answer
At 300 K, $\hbar\omega/k_BT=142$ and $\Theta/k_BT\approx142\,e^{-142}$ , effectively zero. At 4000 K, $\hbar\omega/k_BT=10.6$ and $\Theta/k_BT\approx2.6\times10^{-4}$ , still small, but in the near field the mode density at this frequency is larger by orders of magnitude, and only the product can make a peak visible. As with gold in chapter 2, the surface mode exists; what is missing is thermal energy.
Why is Eq. (7), not Eq. (5), the turning point of the letter?
Show the answer
Eq. (5) is the calculation; it tells you the numbers. Eq. (7) is the explanation; it tells you why. It gives the near-field spectrum the same structure as the Planck spectrum, with the mode energy depending only on temperature and frequency, and puts all the change into the local density of modes. After that step the changes in Fig. 1 can be understood as modes gained and lost, and Eq. (8) and Fig. 3 both analyse that mode density.

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.