L2 · Chapter 2

Evanescent waves and the surface phonon polariton of silicon carbide

Sandbox Physics · 2026-10-30

Part of the field near a hot surface never reaches the far zone. Its in-plane wavevector is too large, so it decays exponentially away from the surface and a distant detector never sees it. On a polar crystal such as silicon carbide, these clinging waves include a true eigenmode, the surface phonon polariton. Its frequency is pinned almost to a single value, and that value is where the bright line of the next chapter’s near-field spectrum sits.

Waves with too large an in-plane wavevector cannot leave

In vacuum a plane wave of frequency $\omega$ has wavenumber $k_0=\omega/c$ . Split the wavevector into a component $\beta$ parallel to the surface and a normal component $k_z$ , so that $\beta^2+k_z^2=k_0^2$ . As long as $\beta<k_0$ , $k_z$ is real and the wave can travel away.

Thermal currents inside the material produce every $\beta$ . When $\beta>k_0$ , $k_z$ becomes imaginary: the field on the vacuum side decays as $e^{-\kappa z}$ and its energy density as $e^{-2\kappa z}$ . This is an evanescent wave. Its period along the surface is shorter than the vacuum wavelength, it weakens with distance, and a distant spectrometer cannot detect it.

Total internal reflection in glass is the same thing: there is an evanescent layer outside the reflecting face, with nothing there to catch it. Push a second block of glass into that layer and light tunnels across. The heat flow of chapters 4 and 5 relies on exactly this tunnelling.

k_z=\sqrt{k_0^{2}-\beta^{2}}=i\kappa,\qquad \kappa=\sqrt{\beta^{2}-k_0^{2}},\qquad |\mathbf E|^{2}\propto e^{-2\kappa z}\quad(\beta>k_0)

Take the branch with

\operatorname{Im}k_z>0

, so the field decays away from the surface.

Source: A. V. Shchegrov · K. Joulain

The pole of the reflection coefficient is the surface wave

Among evanescent waves there is a special solution: with no incoming wave at all, fields on both sides of the interface sustain each other, travel along the surface and decay on both sides. Such a solution needs the denominator of a reflection coefficient to vanish, that is, a pole of the reflection coefficient.

A non-magnetic material has no s-polarised surface wave. The s denominator is $k_{z0}+k_{z1}$ ; when both sides decay, both terms have positive imaginary parts and cannot cancel. The p denominator is $\varepsilon k_{z0}+k_{z1}$ , and with a negative $\varepsilon$ the two terms can cancel.

Insert $k_{z0}^2=k_0^2-\beta^2$ and $k_{z1}^2=\varepsilon k_0^2-\beta^2$ into $\varepsilon^2k_{z0}^2=k_{z1}^2$ and rearrange to get $\beta^2=k_0^2\,\varepsilon/(\varepsilon+1)$ . A real $\beta$ larger than $k_0$ requires $\varepsilon<-1$ .

r_p=\frac{\varepsilon\,k_{z0}-k_{z1}}{\varepsilon\,k_{z0}+k_{z1}},\qquad \varepsilon\,k_{z0}+k_{z1}=0\;\Longrightarrow\;\beta=k_0\sqrt{\frac{\varepsilon}{\varepsilon+1}}

Eqs. (17) and (20) of Joulain et al.; Eq. (6) of Shchegrov et al. Here

k_{z0}

is on the vacuum side and

k_{z1}

inside the material.

Source: K. Joulain · A. V. Shchegrov

Where the dispersion flattens, modes crowd together

The permittivity of SiC is negative between $\omega_T$ and $\omega_L$ . As the frequency rises from $\omega_T$ , $\varepsilon$ climbs from a large negative number toward $-1$ and $\beta/k_0$ grows from just above one to infinity at $\varepsilon=-1$ . Call this frequency $\omega_{\mathrm{sp}}$ .

Near $\omega_{\mathrm{sp}}$ the dispersion curve is almost horizontal: a wide range of $\beta$ maps onto nearly the same frequency. In that narrow band a unit area holds far more surface modes than vacuum holds travelling modes at the same frequency. Joulain and co-workers call it a peak in the density of electromagnetic states.

With the parameters of Shchegrov et al., $\omega_{\mathrm{sp}}=178.74\times10^{12}\,\mathrm{rad\,s^{-1}}$ . Joulain et al. obtain $1.784\times10^{14}\,\mathrm{rad\,s^{-1}}$ from parameters given in wavenumbers; the 0.2% difference comes from rounding in the unit conversion.

When $\beta$ is much larger than $k_0$ , both $k_z$ approach $i\beta$ and the reflection coefficient stops depending on $\beta$ . Its imaginary part depends only on $\varepsilon$ and is largest where $|\varepsilon+1|$ is smallest. The next chapter shows that this imaginary part is the emission strength of each evanescent mode.

\varepsilon(\omega_{\mathrm{sp}})=-1\;\Longrightarrow\;\omega_{\mathrm{sp}}=\sqrt{\frac{\varepsilon_\infty\omega_L^{2}+\omega_T^{2}}{\varepsilon_\infty+1}}

The frequency where the lossless SiC permittivity equals

-1

\beta\gg k_0:\quad r_p\to\frac{\varepsilon-1}{\varepsilon+1},\qquad \operatorname{Im}r_p\to\frac{2\operatorname{Im}\varepsilon}{|\varepsilon+1|^{2}}

The quasi-static limit, where p reflection no longer depends on the wavevector. At 178.7,

\operatorname{Im}\varepsilon/|\varepsilon+1|^2=7.70

; at 166 it is only 0.0097 (in

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

Source: K. Joulain · A. V. Shchegrov

Worked example: how far the surface wave at 166 reaches

Problem. Fig. 3 of Shchegrov et al. shows that at $\omega=166\times10^{12}\,\mathrm{rad\,s^{-1}}$ the energy density decays exponentially between heights of 1 and 5 µm, while at 210 it does not. Find $\beta/k_0$ and the energy decay length $1/(2\kappa)$ of the surface wave at 166, and explain why 210 has no exponential stretch.

Model. Neglect damping and take $\varepsilon(166)=-7.495$ . Find $\beta$ from the dispersion relation, then $\kappa=\sqrt{\beta^2-k_0^2}$ .

Numbers. $k_0=166\times10^{12}/(2.998\times10^{8})=5.537\times10^{5}\,\mathrm{m^{-1}}$ , a vacuum wavelength of 11.35 µm. $\beta^2/k_0^2=7.495/6.495=1.1540$ , so $\beta/k_0=1.074$ . $\kappa/k_0=\sqrt{0.1540}=0.3924$ , $\kappa=2.173\times10^{5}\,\mathrm{m^{-1}}$ and the energy decay length is $1/(2\kappa)=2.30\,\mu\mathrm{m}$ .

Check. The course benchmark script computes the energy density between 2 and 5 µm with the full model, damping and all evanescent waves included, and fits an exponential. It finds 2.286 µm, 0.65% from 2.301 µm. One surface mode really does dominate that stretch.

At 210, $\varepsilon=3.30$ is positive and there is no surface wave. The evanescent energy there comes from a continuous range of $\beta$ with different decay lengths, and their sum is not a single exponential. Closer to $\omega_{\mathrm{sp}}$ the surface wave clings more tightly: at 175, $\beta/k_0=1.35$ and the decay length is 0.95 µm; at 178, $\beta/k_0=2.37$ and 0.39 µm.

Source: A. V. Shchegrov

Try it in the Lab

Compare how two frequencies decay with height

Predict first: between 1 and 5 µm, which of the two frequencies, 166 or 210, gives a straight line on a logarithmic energy axis?
Choose the A line emerges preset and look at the Two frequencies against height chart. Between 1 and 5 µm, measure the height over which the 166 curve falls by a factor of e, and compare with the 2.30 µm of the worked example.
Now look at the Which waves carry the energy map and find the curve the bright band follows. Lower the probe to 100 nm and explain why the band moves to larger in-plane wavevectors.

Why gold shows no line

Predict first: the surface plasmon of gold lies in the near ultraviolet. Will it appear in the spectrum at the probe at 300 K?
Choose the Gold: no line preset, read the strongest frequency, then lower the probe to 10 nm.
Explain with the flat asymptote and the mean mode energy of chapter 1: the surface mode of gold exists, but at 300 K it holds almost no energy.

Let an evanescent wave carry power across a gap

Predict first: beyond the critical angle at a glass–air boundary, does the field in the air carry power away from the surface?
Open the Boundary Wave Tunnel Lab, choose the Single-interface TIR preset, then Two-boundary tunnelling. Narrow the air gap and widen it again, and watch how much power crosses.
Explain with the decay constant of this chapter why one decaying wave carries no power, and why power gets across once a second prism comes close. The tunnelling channels between two plates in chapter 4 are the same effect.

Exercises with explained answers

A material has $\varepsilon=-2$ at some frequency, with no loss. Find $\beta/k_0$ of the surface wave and the ratio of the energy decay length to the vacuum wavelength.
Show the answer
$\beta^2/k_0^2=(-2)/(-1)=2$ , so $\beta/k_0=\sqrt2=1.414$ . Then $\kappa=\sqrt{2k_0^2-k_0^2}=k_0$ and the energy decay length is $1/(2k_0)=\lambda/(4\pi)$ , about 8% of the vacuum wavelength.
Why is there no surface wave when $-1<\varepsilon<0$ ?
Show the answer
Then $\varepsilon/(\varepsilon+1)$ is negative, $\beta^2$ is negative and $\beta$ is not real, so no solution travels along the surface while decaying on both sides. SiC is in this situation between $\omega_{\mathrm{sp}}$ and $\omega_L$ , so its surface wave exists only between $\omega_T$ and $\omega_{\mathrm{sp}}$ .
The plasma frequency of gold is $\omega_P=1.385\times10^{16}\,\mathrm{rad\,s^{-1}}$ . In the Drude model, where is the flat asymptote of its surface mode? What is $\hbar\omega/k_BT$ at 300 K, and how much thermal energy can that mode hold?
Show the answer
The lossless Drude form $\varepsilon=1-\omega_P^2/\omega^2=-1$ gives $\omega_{\mathrm{sp}}=\omega_P/\sqrt2=9.79\times10^{15}\,\mathrm{rad\,s^{-1}}$ . At 300 K, $\hbar\omega/k_BT=249$ and the mean mode energy is about $249\,e^{-249}\,k_BT$ , effectively zero. That is why the near-field spectrum of gold has no matching line.
In the quasi-static limit, compare $\operatorname{Im}r_p$ of SiC at 178.7 and at 166 (in $10^{12}\,\mathrm{rad\,s^{-1}}$ ).
Show the answer
At 178.7, $\varepsilon=-1.008+0.129i$ , $|\varepsilon+1|^2=0.0168$ and $\operatorname{Im}r_p=2\times0.129/0.0168=15.4$ . At 166, $\varepsilon=-7.48+0.41i$ , $|\varepsilon+1|^2=42.2$ and $\operatorname{Im}r_p=0.019$ . The first is about 800 times larger, and the difference is all in the denominator: at 178.7, $\varepsilon$ is almost exactly $-1$ .

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.