Time-resolved interferometric detection of ultrashort strain solitons in sapphire

We study one-dimensional nonlinear propagation of high-amplitude acoustic waves in sapphire, for various sample temperatures, sample thicknesses, and pump fluences. Strain waves are generated in a 100-nm-thick chromium film and launched into the sapphire. For temperatures $<60\text{K}$, damping can be neglected and propagation is dominated by the nonlinear and dispersive properties of the sapphire substrate. An interferometric technique is used to detect the wave on an epitaxially grown $\sim 20\text{-nm}$-thick Cr film at the opposite side of the sample. At the lowest temperature of 18 K, a train of up to seven solitons is detected in sapphire for a pump fluence of $11\text{mJ}/{\text{cm}}^2$. From the soliton amplitudes and velocities, we infer soliton temporal and spatial widths as short as 200 fs and 2 nm. A theoretical analysis based on numerical solution of the Korteweg-de Vries-Burgers equation yields excellent agreement to all experiments presented. Deviations to the direct theoretical result can be explained by pump intensity variations, affecting the (nonlinear) propagation properties.

Figure 1

(Color online) (a) Measured (black line) and calculated (red or gray line) phase signals for room temperature (upper trace), and cooling via 200, 100, 60 to 18 K (bottom trace) for a sapphire sample thickness of $307\mu \text{m}$ and a constant pump fluence of $9.7\text{mJ}/{\text{cm}}^2$. Calculations include broadening effects. Numbers $i=1,\dots ,5$ at 60 K trace correspond to arrival of $i\text{th}$ soliton at the detection surface. (b) Calculated strain wave shape after propagation through $307\mu \text{m}$ of sapphire, given the input strain amplitude $s_0\left(\times {10}^3\right)$ required to reach agreement with the measurements in (a). Inset: sketch of experimental geometry.

Figure 2

(Color online) (a) Measurements (black line) and calculations (red or gray line) of amplitude $\left(\rho \right)$ signal for wave propagation through $123\mu \text{m}$ of sapphire at a temperature of 18 K. Numbers indicate pump fluence in $\text{mJ}/{\text{cm}}^2$. Bipolar features correspond to arrival of solitons. For $8.8\text{mJ}/{\text{cm}}^2$, soliton arrival times are indicated by arrows. In the calculation, corrections for temporal broadening discussed in Sec. 5 were included. At fluence of $8.8\text{mJ}/{\text{cm}}^2$, both uncorrected and corrected signals are shown, demonstrating the relevance of temporal broadening effects. (b) Calculated strain profiles after propagation for an input pulse of width $\sim 3.6\text{ps}$ and indicated strain amplitudes $s_0$ $\left(\times {10}^3\right)$. At the highest fluence, at least five solitons are formed. Note that the final solitons in the calculation are not resolved in the measurement.

Figure 3

(Color online) (a) Measured (black line) and calculated (red or gray line) phase $\left(\delta \phi \right)$ signals for the $123\text{−}\mu \text{m}$-thick sample at 18 K. Numbers indicate fluence in millijoule per square centimeter. Steps correspond to soliton arrival at the detection surface. In the calculation, broadening effects discussed in Sec. 5 are taken into account. (b) Calculated strain wave form after propagation through $123\mu \text{m}$ of sapphire. Numbers indicate input strain amplitude $s_0\times {10}^3$.

Figure 4

(Color online) (a) Measured (black line) and calculated (red or gray line) $\delta \phi$ signals for the $307\text{−}\mu \text{m}$-thick sample at 18 K. Numbers indicate fluence in millijoule per square centimeter. In the calculation, broadening effects discussed in Sec. 5 are taken into account. (b) Calculated strain wave form after propagation through $307\mu \text{m}$ of sapphire. Numbers indicate input strain amplitude $s_0\times {10}^3$.

Figure 5

(Color online) (a) Measured (black line) and calculated (red or gray line) $\delta \phi$ signals for the $405\text{−}\mu \text{m}$-thick sample at 18 K. Numbers indicate fluence in millijoule per square centimeter. In the calculation, broadening effects discussed in Sec. 5 are taken into account. Calculated data for fluence of $9.8\text{mJ}/{\text{cm}}^2$ shows difference between broadened (solid line) and unbroadened (dashed line) signals. (b) Calculated strain wave form after propagation through $405\mu \text{m}$ of sapphire. Numbers indicate input $s_0\times {10}^3$. (c) Detail of solitons in measured trace at $5.9\text{mJ}/{\text{cm}}^2$, and calculation with and without metal film contribution (red solid and blue dashed line, respectively). The width $w_1$ of the first soliton was measured at $\sim 500\text{fs}$. (d) $\sim 500\text{GHz}$ oscillations in the trailing part of the wave for $5.9\text{mJ}/{\text{cm}}^2$ and corresponding calculated result (red line).

Figure 6

(Color online) (a) Symbols: fitted strain amplitude $s_0$ versus pump fluence at 18 K. Fitted temporal width was 3.6 ps in all cases. Lines: values for $s_0$ derived from maximum phase signal amplitudes. (b) Temporal width determined for the first three soliton signals (squares, circles, and triangles, respectively) and sample thickness $405\mu \text{m}$. Lines are expected nonlinear broadening $\sim 0.07\delta t_i$ based on measured arrival times $t_i$.

Figure 7

(a) Position of soliton versus sample thickness for an initial strain wave of amplitude $s_0=4.3\times {10}^{-4}$ and width $\tau =3.6\text{ps}$. Numbers indicate soliton velocity in meter per second. (b) Like (a) but now for $s_0=1.6\times {10}^{-3}$. All velocities are determined $\pm 0.1\text{m}/\text{s}$. No accurate determination of velocity could be performed for the sixth soliton.

Figure 8

(a) Position of solitons versus similarity parameter ${\sigma }_N$. Dashed lines give comparison to arrival times predicted by Eqs. (7, 10, 13). (b) Soliton volume determined from measurements versus similarity parameter ${\sigma }_N$. Dashed lines are soliton volume $V_i=2w_i{a}_i$ predicted by theory.