Subharmonic Resonant Optical Excitation of Confined Acoustic Modes in a Free-Standing Semiconductor Membrane at GHz Frequencies with a High-Repetition-Rate Femtosecond Laser

We propose subharmonic resonant optical excitation with femtosecond lasers as a new method for the characterization of phononic and nanomechanical systems in the gigahertz to terahertz frequency range. This method is applied for the investigation of confined acoustic modes in a free-standing semiconductor membrane. By tuning the repetition rate of a femtosecond laser through a subharmonic of a mechanical resonance we amplify the mechanical amplitude, directly measure the linewidth with megahertz resolution, infer the lifetime of the coherently excited vibrational states, accurately determine the system’s quality factor, and determine the amplitude of the mechanical motion with femtometer resolution.

Figure 1

Time-resolved modulation of the reflected probe beam $\Delta R/R_0$ measured with a repetition rate $f_R=800\mathrm{MHz}$. The inset shows these oscillations after subtracting the electronic contribution. (Note the change of 3 orders of magnitude in the $y$ axis between the main figure and the inset.)

Figure 2

(a) Fourier spectra of the transients recorded with two different repetition rates. The curve for $f_R=800\mathrm{MHz}$ represents the out-of-resonance situation, while the other curve represents the exact resonant condition ($f_R=999.99\mathrm{MHz}$). Odd confined modes are labeled with their mode index ($n=1,3,\dots 15$). (b) Resonant behavior of the amplitude of fundamental mode $f_n=19\mathrm{GHz}$ ($n=1$) when $f_R$ is varied as indicated.

Figure 3

Typical transients for different pump beam repetition rates ($f_R$) as indicated. $f_R=800\mathrm{MHz}$ corresponds to the far-from-resonance situation (also shown in Fig. 1), and $f_R=999.99\mathrm{MHz}$ to the exact resonant condition with the 18th subharmonic of the membrane’s fundamental confined acoustic mode $f_1=19\mathrm{GHz}$. The transients are shown vertically shifted for clarity.

Figure 4

Oscillations corresponding to the membrane’s confined longitudinal acoustic modes for different excitation repetition rates $f_R$. (a) Experimental oscillations, derived from Fig. 3. (b) Corresponding simulated transients. The color legends correspond to the one shown in Fig. 3.

Figure 5

(a) Phase of the extracted oscillations with respect to the resonant condition, and (b) normalized spectral amplitude of the $f_1$ acoustic confined mode as function of the laser repetition rate $f_R$. The full lines are the result from the simulations using the theoretical model described in the text. The arrows indicate the corresponding $f_R$ positions of the transients and Fourier spectra shown in Figs. 2, 3, 4 for the respective colors. The top axis indicates the frequency of the driven $\Delta R/R_0$ oscillations, i.e., 19 times $f_R$.