L2 · Chapter 4

Transient thermal gratings: measuring heat diffusion with interference fringes

Sandbox Physics · 2026-10-30

To measure heat transport over a few micrometres you do not need a microscopic heater and thermometer. Cross two short laser pulses on a sample, and the absorbed interference fringes become a row of hot and cold stripes. As heat spreads the stripes flatten, and a third beam diffracting off them reads how fast. The method is called the transient thermal grating, and its period is set by the crossing angle, so it can be tuned continuously. This chapter shows how the decay time gives the diffusivity, and why the measured diffusivity drops as the period shrinks.

Painting hot and cold stripes with two beams

Two pump beams of wavelength $\lambda$ crossing at an angle $\theta$ make sinusoidal interference fringes on the sample, with the period given by Maznev and colleagues (below). The sample absorbs light in the bright fringes and not in the dark ones, so the temperature varies along the surface as $\cos(qx)$ , with $q=2\pi/L$ .

For graphite, Huberman and colleagues pumped with 515 nm pulses and probed with a continuous 532 nm beam. Thermal expansion turns the temperature stripes into a surface ripple, which acts as a reflection grating and diffracts the probe. Mixing the diffracted light with a reference beam, heterodyne detection, gives a signal proportional to the ripple amplitude $u(t)$ with its sign: if hot and cold stripes swap places, the signal turns negative.

Graphite conducts about 300 times less heat across its layers than along them, so until the stripes are washed out the heat moves almost only along the surface. The ripple is proportional to the temperature rise integrated over depth, and the signal reports one-dimensional in-plane transport alone.

L=\frac{\lambda}{2\sin(\theta/2)}

A wider crossing angle gives a shorter period. The period is fixed by the optics, not by the sample.

Two pump beams of wavelength

\lambda

crossing at angle

\theta

interfere on the surface and leave temperature stripes with period

L=\lambda/(2\sin(\theta/2))

. A probe beam diffracts from the surface ripple that the stripes cause. Under diffusion the stripe amplitude decays as

e^{-t/\tau}

with

\tau=L^2/(4\pi^2\alpha)

. Original schematic, not to scale.

Source: A. A. Maznev · S. Huberman

Diffusion makes a sinusoidal grating decay exponentially

Substitute $T=\theta(t)\cos(qx)$ into the one-dimensional diffusion equation. The second derivative of a cosine is the cosine times $-q^2$ , so the stripes keep their shape and only the amplitude decays, following a first-order ordinary differential equation.

That is the beauty of the method: a sinusoid is an eigensolution of the diffusion equation, so you need neither the edges of the heated spot nor the spatial profile; one time constant says it all. Huberman and colleagues write the same time constant as below.

\frac{d\theta}{dt}=-\alpha q^{2}\theta\quad\Longrightarrow\quad\theta(t)=\theta(0)\,e^{-t/\tau_d},\qquad \tau_d=\frac{1}{\alpha q^{2}}=\frac{L^{2}}{4\pi^{2}\alpha}

Turned around, a measured decay time gives the apparent diffusivity

\alpha_{\mathrm{app}}=L^2/(4\pi^2\tau_d)

Source: S. Huberman · A. A. Maznev

At short periods the decay is slower than diffusion

Huberman and colleagues measured periods from 3.7 to 37.5 µm at 300 K. The long period gave a diffusivity of 11 cm²/s. At shorter periods the apparent diffusivity $L^2/(4\pi^2\tau_d)$ was no longer constant: the decay was slower than the diffusion equation predicts, while the traces stayed exponential. They read this as the onset of a size effect, once the mean free paths of heat-carrying phonons become comparable to the period.

Maznev and colleagues explained the slowdown theoretically. Low-frequency phonons with long mean free paths fly from a hot stripe straight into cold ones several stripes away, so they do not smooth the pattern the way diffusion would; in their analysis these phonons contribute less to the grating decay than to bulk conduction. For silicon at room temperature they found the drop in effective conductivity still significant at periods as large as 10 µm.

The gray model has an extreme case: with no scattering at all the amplitude follows $\sin(qvt)/(qvt)$ , and the decay time scales with the period instead of its square. The Lab’s “Free flight” preset shows this case.

Source: S. Huberman · A. A. Maznev

Fitting and held-out testing

Each record in the Lab is a model curve plus seeded Gaussian noise. An exponential fit gives a decay rate and its standard error, which convert to an apparent diffusivity. Huberman and colleagues did something similar, fitting the raw data from the moment the signal had fallen to 80% of its maximum.

The Lab sweeps seven periods at once. Five are used to fit two candidate models: a Fourier model with one diffusivity for all periods, and the damped-wave model of the next chapter. The other two periods are held out. A model explains the data only if it predicts the held-out records to within the noise. This is stricter than a good fit, because a model with more parameters always fits its training data more easily. A heavily damped wave is just diffusion, so a passing damped wave does not by itself mean second sound; the Lab also counts at how many swept periods the fitted wave really swings below zero.

Source: S. Huberman · A. A. Maznev

Worked example: time scales in graphite and silicon

Problem: Huberman and colleagues pumped at 515 nm and found a graphite diffusivity of 11 cm²/s at a 37.5 µm period and 300 K. Find the crossing angle and decay time for that period; then the crossing angle at 3.7 µm and the decay time Fourier’s law predicts there. Finally, for gray silicon ( $D=0.797\,\mathrm{cm^2/s}$ ), find the decay times at 1 µm and 10 µm.

At 37.5 µm, $\sin(\theta/2)=0.515/(2\times37.5)$ , so $\theta=0.787^\circ$ . The decay time is $\tau_d=(37.5\,\mu\mathrm m)^2/(4\pi^2\times11\,\mathrm{cm^2/s})=32.4\,\mathrm{ns}$ .

At 3.7 µm, $\theta=7.98^\circ$ and Fourier’s law predicts $\tau_d=0.315\,\mathrm{ns}$ . A period about ten times shorter gives a time about a hundred times shorter. The measured decay is slower, so the apparent diffusivity comes out below 11 cm²/s.

Gray silicon: $\tau_d=0.318\,\mathrm{ns}$ at 1 µm and $31.8\,\mathrm{ns}$ at 10 µm, in line with the “about 0.3 to 30 ns for periods of 1 to 10 µm” quoted by Maznev and colleagues.

Source: S. Huberman · A. A. Maznev · Ioffe Institute

Try it in the Lab

From plain diffusion to short periods

Choose “Plain diffusion”. Check that the record follows the blue Fourier dashes, and read the diffusivity from an exponential fit over the Fourier value.
Choose “Short period: decay slows”. Predict whether the ratio will be above or below 1, then read it.
In the middle chart, look at the fitted diffusivity across seven periods and find where it starts to leave the Fourier value.

Drag the stage to change the period

Drag the stage sideways from long to short periods and watch the pump crossing angle on the stage change.
Choose “Long period: Fourier returns” and confirm that the ratio is back near 1.

Exercises with explained answers

What crossing angle gives a 10 µm period with a 515 nm pump?
Show the answer
$\theta=2\arcsin\bigl(0.515/(2\times10)\bigr)=2.95^\circ$ . The Lab lists the same angle among its detailed numbers.
Gray silicon decays in 31.8 ns at a 10 µm period. What does Fourier’s law give if the period is halved?
Show the answer
The decay time scales with the square of the period, so halving the period quarters the time, to about 7.94 ns.
Suppose graphite decays in 0.40 ns at a 3.7 µm period (a number assumed for this exercise). What is the apparent diffusivity, and what fraction of the long-period value is it?
Show the answer
$\alpha_{\mathrm{app}}=(3.7\times10^{-6})^2/(4\pi^2\times0.40\times10^{-9})\,\mathrm{m^2/s}=8.67\,\mathrm{cm^2/s}$ , which is $0.788$ of 11 cm²/s.
Can a thermal grating signal go negative during diffusion?
Show the answer
No. The diffusion solution $\theta(0)e^{-t/\tau_d}$ always keeps the sign it started with. As Huberman and colleagues put it, in diffusion heat only flows from hot to cold, so hot and cold stripes cannot swap places. A negative signal is a mark of non-diffusive transport, the subject of the next chapter.

Sources and what we recomputed

Ioffe Institute, NSM Archive: Thermal properties of Silicon (Si) and Mechanical properties, elastic constants, lattice vibrations of Silicon (Si)
Both pages read in full (the mechanical page is at mechanic.html in the same folder). Values used at 300 K: specific heat 0.7 J/(g K), thermal conductivity 1.3 W/(cm K), thermal diffusivity 0.8 cm²/s, density 2.329 g/cm³, and the sound speeds 8.43 km/s (longitudinal) and 5.84 km/s (transverse) along the [100] direction. The course combines the two speeds into one Debye average; that average is our step, not a value from the pages.
A. A. Maznev, J. A. Johnson and K. A. Nelson, Onset of nondiffusive phonon transport in transient thermal grating decay, Physical Review B 84, 195206 (2011), arXiv:1108.3770v2
Read in full (arXiv v2). Introduction: the kinetic-theory mean free path of tens of nanometres, the gray-body approximation as an oversimplification, the slab between two black walls favoured by theorists, the grating period set by the crossing angle of the pump beams, and grating decay times of about 0.3 to 30 ns in silicon for periods of 1 to 10 µm. The abstract states that the reduction of the effective conductivity in silicon is significant at periods as large as 10 µm. The course quotes no equation numbers from this paper.
S. Huberman, R. A. Duncan, K. Chen, B. Song, V. Chiloyan, Z. Ding, A. A. Maznev, G. Chen and K. A. Nelson, Observation of second sound in graphite at temperatures above 100 K, Science (2019), doi:10.1126/science.aav3548, arXiv:1901.09160v1
Read in full from arXiv v1, including the supplementary materials. Used here: the transient-grating setup with a 515 nm pump and a 532 nm probe in heterodyne detection; the diffusion time constant written as the period squared over four pi squared times the diffusivity; 11 cm²/s at a 37.5 µm period at 300 K and the slower-than-diffusive decay at short periods (Fig. 1); the sign flip at 85 K and the velocity of about 3200 m/s from the first minimum (Fig. 2); oscillations at 104 K and 125 K that vanish at 150 K and at 50 K (Fig. 3); the predicted window between 50 and 250 K (Fig. 4); the first-principles value of 3650 m/s; and, from the supplement, the dip minimum near 1.2 ns for the 7.5 µm trace and the grating amplitude of about 0.2 K during the oscillation. The published Science version was not compared line by line with arXiv v1.