Picosecond time resolved opto-acoustic imaging with 48 MHz frequency resolution.

A compact femtosecond dual-oscillator pump-probe setup with 48 MHz-repetition rate, relying on asynchronous optical sampling, is presented. The relative timing jitter between both lasers over the whole pump-probe delay range is of the order of or lower than 500 fs. We demonstrate that both a picosecond temporal resolution and a 48 MHz spectral resolution combined with the fast acquisition rate inherent to the asynchronous optical sampling allow performing broadband opto-acoustic imaging with a spectrum covering more than two decades from 300 MHz to 150 GHz. As an illustration, the opto-acoustic response of a supported thin film is investigated, revealing high frequency acoustic echoes close to the epicenter as well as low GHz surface acoustic waves propagating up to 40μm away from the epicenter. Semi-analytical calculations have been carried out and perfectly reproduce the dispersion of the surface acoustic waves experimentally observed.


Introduction
The understanding and control of elastic properties of matter at a nanometer scale as well as the investigation of the vibrational landscape of nanostructures in the GHz to THz range has considerably made progress since the emergence of the picosecond opto-acoustics in the mid 80's [1,2]. In its original form, this experimental method relies on a conventional femtosecond pump-probe setup in a transient reflectivity configuration. Pump and probe pulses are delivered by the same laser and the probe pulse is time delayed with respect to the pump pulse by using moving mirrors mounted on a motorized translation stage. This mechanical motion shows several intrinsic drawbacks, such as fluctuation of the beam pointing, modification of the size of the focal spot on the sample and long acquisition times for pump-probe delays ranging over several nanoseconds [3]. To circumvent these drawbacks, Elzinga et al. have introduced asynchronous optical sampling (ASOPS) in 1987 [4]. In this scheme, the moving mirrors are no more necessary and the pump-probe delay is generated by using two pulsed lasers at slightly different repetition rates. The use of the ASOPS technique led to fundamental breakthrough in Fourier transform dual-comb spectroscopy [5], THz time-domain spectroscopy [6][7][8], nanoscale thermal transport [9-11]. The combined use of ASOPS with picosecond acoustics has emerged from the development of the dual-GHz femtosecond oscillator initiated by Janke et al. [3,12,13]. This high repetition rate is ideally suited for investigating the propagation of coherent acoustic phonons with frequencies ranging from few tens of GHz [14] to several THz [15]. In the case of confined acoustic phonons with long lifetimes, the intrinsic spectral resolution imposed by the laser repetition rate can be overcome with a subharmonic resonant optical excitation [16]. However, the downside of such ASOPS setups with high repetition rates is that they do not offer a sufficient spectral resolution to investigate the propagation of low GHz coherent acoustic phonons. For instance, although high frequency surface acoustic waves (SAWs) have been generated by using one dimensional [17,18] or two dimensional [19] metallic lithographed arrays, the spectrum of SAWs generated through the optical absorption of a focused laser pulse is in general limited by the optical diffraction limit. Within such limitation, this spectrum ranges from few hundreds of MHz to few GHz [20][21][22][23][24][25]. Up to now, in the picosecond acoustics field, ASOPS pump-probe setups with low repetition rate lasers [6,8,[26][27][28] have taken benefit from the high temporal resolution of the system for investigating the propagation of sub-THz acoustic phonons [29,30].
Another advantage of ASOPS setups is the drastic reduction of acquisition times. Hettich et al. have taken benefit of this high acquisition rate to perform time-resolved opto-acoustic imaging over a narrowband acoustic spectrum centered around 80 GHz [31], as it can also be achieved with conventional pump-probe setups [21,24,[32][33][34]. However, the combination of both high spectral and high temporal resolutions has not yet been exploited to image the propagation of coherent acoustic phonons.
In this article we present an ASOPS-based setup with repetition rates close to 48 MHz and we demonstrate that this system is ideally suited to perform broadband opto-acoustic imaging with a 1 ps-temporal resolution and a 48 MHz-spectral resolution. In the first part, we describe the structure of the dual-oscillator and we quantify the temporal resolution of the ASOPS pump-probe setup. The latter is in part imposed by the relative timing jitter between both femtosecond laser combs which turns out to be of the order of or lower than 500 fs over the whole pump-probe delay range. In the second part, the ability of this setup to perform broadband opto-acoustic imaging with acoustic spectrum covering more than two decades is illustrated by investigating the propagation of the acoustic waves generated by an opto-acoustic point source located at the free surface of a supported thin film. The upper part of the spectrum, up to 150 GHz, corresponds to bulk acoustic waves (BAWs) propagating back and forth inside the thin film and allows to accurately deduce the thin film thickness. In the lower part of the spectrum, close to 1 GHz, two surface acoustic waves (SAWs) are detected, namely a surface skimming longitudinal wave (SSLW) and a pseudo Rayleigh wave (PRW). The strong dispersion of the low frequency PRW is then analyzed by carrying out semi-analytical calculations. The film thickness is a key parameter for these calculations and it has been measured from the high frequency acoustic echoes. An excellent agreement between experimental and calculated results is demonstrated. The 48-MHz spectral resolution allows to illustrate the dispersion of the PRW in the spectral domain by analyzing the frequency dependence of the group velocity in the 0.3 -1.2 GHz range.

Structure of the dual-oscillator
The laser source is a compact dual-oscillator (t-Pulse Duo, Amplitudes Systèmes, France) [35]. This system contains two diode-pumped passively mode-locked Yb:KYW laser cavities. The two cavities are located on the same temperature-regulated breadboard. A schematic of the dual-oscillator structure is given in Fig. 1. The asynchronous optical sampling consists in synchronizing the repetition rate of a slave laser (laser #2) at a slightly different value compared with the repetition rate of the master laser (laser #1). In the present dual-oscillator system, the master laser delivers 1.0 W average power, 330 fs pulses at 1027 nm and at a repetition rate f rep,1 . The slave laser delivers 0.6 W average power, 430 fs pulses at 1040 nm and at a repetition rate f rep,2 . The repetition rate of each laser is approximately 48 MHz. The beating frequency Δ f between the two lasers is generated by a frequency synthesizer and is stabilized using an active feedback [36]. Two fast photodiodes PD1 and PD2, followed by low-pass filters (LPF), monitor the fundamental repetition rate of each laser. The electronic signal at f rep,1 delivered by PD1 is upshifted by the frequency synthesizer at a slightly higher value f rep,1 + Δ f , where Δ f = 500 Hz.
A phase detector compares f rep,2 and f rep,1 + Δ f . One of the cavity mirrors of the slave laser is mounted on a piezoelectric transducer (PZT). The amplified output of the phase detector drives the control voltage of the PZT. The slave cavity length is controlled by the PZT voltage and is adjusted to fulfill the condition f rep,2 = f rep,1 + Δ f . In the following, the slave (resp. the master) laser will be used as the pump (resp. the probe) laser.

HV amplifier
Laser #1 (Master) Simplified scheme of the dual-oscillator system. The fast photodiode PD1 (resp. PD2), associated to the low-pass filter LPF1 (resp. LPF2), monitors the fundamental repetition rate f rep,1 (resp. f rep,2 ) of the master laser (resp. slave laser). The frequency synthesizer generates an up-shifted frequency f rep,1 + Δ f . The amplified phase detector output controls the piezoelectric transducer (PZT) voltage to adjust the slave cavity length in order to fulfill the condition f rep,2 = f rep,1 + Δ f .

Temporal resolution
The time resolution Δτ of the asynchronous pump-probe setup is imposed by four independent factors [10, 13]: (i) the pump-probe delay increment Δτ i = Δ f /( f rep,1 × f rep,2 ) generated by the beating frequency Δ f , (ii) the integrated response of several probe pulses over Δτ d = f /(B × f rep,2 ) since the detector bandwidth B is lower than the lasers repetition rate, (iii) the laser pulses duration Δτ 1 and Δτ 2 and (iv) the relative timing jitter Δτ j between the two synchronized lasers.
With Δ f = 500 Hz and f rep,1 ≈ f rep,2 ≈ 48 MHz, Δτ i is measured at 225 fs. The detector bandwidth of B = 15 MHz leads to Δτ d = 700 fs. The pulses duration Δτ 1 and Δτ 2 and the jitter Δτ j are measured with a home-built all-optical correlator presented in Fig. 2. In order to measure temporal duration of the pulses, each laser is first injected separately and its auto-correlation is recorded. Assuming a squared-hyperbolic secant temporal profile, probe and pump pulses durations are measured at Δτ 1 = 430 fs and Δτ 2 = 330 fs, respectively. The relative timing jitter characterization is then performed by injecting the two lasers simultaneously in the correlator and by acquiring the resulting cross-correlation between pump and probe pulses. This acquisition scheme relies on the idea developed by Gebs et al. [15]. The optical path difference ΔL selects the pump-probe delay at which the jitter is measured. In order to cover the whole pumpprobe delays offered by the lasers repetition rate, i.e. from 0 to 21 ns, ΔL has to be adjustable from 0 to 6.3 meters. Figure 3 shows the relative timing jitter for Δ f = 500 Hz and for pump-probe delays ranging from 0 to 21 ns. The jitter increases with respect to the pump-probe delay, as expected for passively mode-locked lasers [37]. It is worthwhile noting that, in spite of this cumulative effect, the jitter Δτ j is of the order of or lower than 500 fs even for delay as large as 21 ns. This value is close to the jitter reported by Yasui et al. with two synchronized 50.5 MHz modelocked femtosecond lasers [8]. The overall time resolution Δτ in the pump-probe experiments presented in the following is given by the quadratic sum of Δτ 1 , Δτ 2 , Δτ j , Δτ i and Δτ d . Δτ is therefore of the order of 1 ps.

Asynchronous pump-probe opto-acoustic imaging setup
The time-resolved opto-acoustic imaging setup relies on the pump-probe scheme in a transient reflectivity configuration presented in Fig. 4(a). All the results presented hereinafter have been obtained for a beating frequency Δ f = 500 Hz. The two lasers are recombined by the beamsplitter BS4 and focused on the sample through a ×50 microscope objective. The reflected probe beam is directed with the beamsplitter BS5 towards a amplified silicon photodiode (PD). Pump and probe pulse durations are 430 fs and 330 fs, respectively: these durations are sufficiently long to neglect the temporal broadening induced by the dispersion of the different optical elements throughout the setup. A part of each laser is picked off (BS1 and BS2), recombined (BS3) and focused through a ×10 microscope objective on a GaAsP two-photon absorption photodiode (TPA-PD). The TPA-PD signal is amplified by a custom-built preamplifier. At repeated times delayed of the beating period 1/Δ f , the transient electric signal delivered by the TPA-PD is used as a trigger signal for the transient reflectivity acquisition. To perform a two dimensional scan, the pump-probe distance at the surface of the sample is controlled with an afocal telescope lens pair (L1 and L2, f = 100 mm focal lengths) inserted in the pump beam before the beamsplitter BS4. The lens L2 is fixed and the lens L1 is mounted on a motorized xystage. This controlled translation of L1 imposes the incident angle of the collimated pump beam on the entrance pupil of the microscope objective and thus controls the pump-probe distance on the sample surface. By varying this angle, a two dimensional scan over a 100 µm × 100 µm area can be performed. At each pixel of the image, the transient reflectivity is recorded over 21 ns with a subpicosecond temporal step. The acquisition time of the pump-probe signal for each pixel varies from few seconds to few tens of seconds, depending on the signal-to-noise ratio of the measurements. In other words, the strength of the ASOPS rests in its ability to acquire several thousands of temporally resolved images of the transient reflectivity. This inherent consequence of the ASOPS turns out to be one of the main advantages compared to conventional methods implying the use of a mechanical delay line [21, 24].

Opto-acoustic response of a supported thin film
The sample under investigation is depicted in Fig. 4(b). A thin tungsten layer is deposited on top of a (100)-silicon substrate. The pump pulse is focused on the free surface of the film. The energy of the pump pulse at the sample position is 0.5 nJ. Following the absorption of the pump pulse, the thermo-elastic coupling leads to the generation of several acoustic waves over a broad spectral range from approximately 100 MHz to 100 GHz [20]. Notably, a longitudinal acoustic wave is launched in the z-direction. Due to the acoustic impedance mismatch at the W-Si interface, this wave bounces back and forth in the thin film and gives rise to acoustic echoes at the tungsten free surface. These echoes are detected by the probe pulse through the photoelastic interaction over the optical penetration depth in tungsten. The energy of the probe pulse at the sample position is 0.1 nJ. Following the pump pulse absorption, the thermo-elastic coupling leads also to the generation of GHz surface acoustic waves (SAWs) propagating in the xy-plane. The GHz frequency content of the SAWs is imposed by the lateral size (≈ 1.5µm) of the focus pump spot on the free surface of the film. The xy in-plane derivative of the normal displacement of the surface film is detected through a deflectometry detection scheme [38][39][40][41]. Figure 5 shows a raw experimental transient reflectivity at the epicenter over the first 10 ns. Following the coincidence, high frequency acoustic echoes (spikes just after the coincidence) and low frequency surface acoustic waves (up to ≈ 1.5ns) are clearly visible, as well as a slow thermal diffusion background. The absolute magnitude of the acoustic signals (echoes and SAWs) ranges from a few 10 −5 to a few 10 −4 . Data shown in Fig. 5 have been acquired in 10 s and result from the averaging of 5000 acquisitions, leading to a signal resolution of 3 × 10 −6 . High repetition rate ASOPS system present the advantage of a low acquisition time at the expense of the frequency resolution [3]. We have opted for lower repetition in order to get a higher spectral resolution at the expense of the acquisition time. In the following, the emphasis will be put on how the combination of a picosecond temporal resolution and a 48 MHz-spectral resolution allows to dynamically image at the same time 100 GHz BAWs and low GHz SAWs. This broadband acoustic imaging will give a thorough understanding on the propagation of SAWs at the surface of a supported thin film.

Broadband opto-acoustic imaging
The acoustic echoes in the thin film contribute significatively to the transient reflectivity only a few microns away from the epicenter in the xy plane. Figure 6 shows 8 × 8 µm 2 snapshots of the transient reflectivity centered on the epicenter and extracted at four pump-probe delays τ (81 ps, 92 ps, 99 ps and 107 ps). The thermal diffusion background has been subtracted in the data presented in Fig. 6. The values of ΔR/R, displayed on a color scale, directly bring information on the acoustic contribution. ΔR/R = 0 corresponds to the red level. The spatially resolved acoustic echo is clearly visible with a ΔR/R minimum at τ = 92 ps and maximum at τ = 99 ps. These four snapshots demonstrate that dynamical imaging on a picosecond timescale can be achieved with the asynchronous pump-probe setup presented above. While BAWs are confined close to the epicenter, SAWs propagate over tens of microns away from the opto-acoustic source location. Figure 7 shows 45 × 45 µm 2 snapshots of the transient reflectivity extracted at four pump-probe delays τ (2900 ps, 5000 ps, 8000 ps and 14000 ps). Raw images consist of 300 pixels distributed over a 45µm × 45µm surface. To obtain higher spatial resolution for the snapshots displayed in Fig. 7, we have taken benefit from the propagative feature of the acoustic waves. It implies that temporal and spatial scales are intrinsically related via the local instantaneous velocity of the acoustic waves. For each pixel at one given delay, a windowed time correlation between neighboring pixels allows to extract the group velocity as well as the propagation direction and thus to reconstruct the missing pixels. This approach is very robust to dispersion since we exploit the picosecond temporal resolution allowed by the ASOPS system (20,000 frames of 18x18 pixels over 20 ns). The velocity is thus recalculated for each pixel and each of the 20000 temporal steps. With this method, we can reach in each direction a 10-fold increase of the spatial resolution. x and y values denote the position relative to the epicenter. At such pump-probe delays and far away from the epicenter, the thermal contribution becomes negligible and the transient reflectivity, displayed on a color scale, carries information only on the acoustic contribution. ΔR/R = 0 corresponds to the orange level. The signal in the upper-right corner of the snapshot at τ = 2900 ps corresponds to SAWs generated by the previous pump pulse. At τ = 8000 ps, two SAWs are clearly distinguished. The fastest SAW, in the upper-right corner of the snapshot, is almost not dispersive and is a surface skimming longitudinal wave (SSLW). The slowest and strongly dispersive SAW, as highlighted in the snapshot at τ = 14000 ps, corresponds to a pseudo Rayleigh wave (PRW) [33,42]. A more detailed analysis of the dispersion of the PRW will be carried out in section 3.5. Therefore, the 48 MHz-repetition rate of the lasers allows to retrieve information on the SAWs propagation over long temporal delays, or in other words with a high spectral resolution. to τ = 100 ps is the acoustic echo resulting from the first round-trip of the high frequency BAW inside the thin film. The spatially resolved transient reflectivity of this echo has been presented in Fig. 6. Following this first echo, similar transient features with decreasing amplitude are detected at regular time intervals and correspond to several round-trips of the BAW inside the thin film. We assume bulk elastic parameters for the tungsten film [43]: c 11 = 523 GPa and c 44 = 161 GPa. A film thickness e = 252 nm is thus deduced from the temporal delay Δt = 97 ps between two successive echoes. Figure 8(b) shows the spectrum of the transient reflectivity over the first hundreds of picoseconds displayed in Fig. 8(a) and illustrate that the frequency content of the detected acoustic echoes extends up to ≈ 150 GHz.

Low frequency surface acoustic waves -Analysis of the SAWs dispersion
Despite acoustic anisotropy of silicon, acoustic wavefronts of the SAWs presented in Fig. 7 are almost perfectly circular in the xy-plane. This observation differs from results reported on SAWs propagating on a silicon substrate covered with a 50 nm thick gold film [32]. However, in the present case, the tungsten film is five time thicker and make the SAWs almost insensitive to the acoustic anisotropy of silicon. Therefore, in the following, the SAWs propagation will be analyzed along one radius from the epicenter. The chosen radius is defined by y = 0 and x > 0. Figure 9 shows the dynamics of the transient reflectivity over 19 ns for 14 distances from the epicenter ranging from 2.8t o3 8 .4 µm. Continuous black curves correspond to experimental data. All the curves have been vertically upshifted for the sake of clarity. These results exhibit two distinct SAWs, the slower being strongly dispersive.  Fig. 9. Dynamics of the xy in-plane derivatives of the normal displacement of the surface film for 14 pump-probe distances. The value of each pump-probe distance is indicated in blue above each curve. Continuous black curves (resp. red dotted curves) correspond to experimental (resp. calculated) normalized signals. All curves have been vertically upshifted for the sake of clarity. The fastest SAW is a surface skimming longitudinal wave (SSLW) and the slowest and strongly dispersive SAW is a pseudo Rayleigh wave (PRW).
To identify more precisely these two waves, a semi-analytical calculation has been carried out. The underlying model describes the thermo-elastic generation of both volume and surface acoustic waves by a point source located at the free surface of a supported thin film [44,45]. The calculated deflectometric signal, i.e. dynamics of the xy in-plane derivatives of the normal displacement of the surface film [38][39][40][41], are plotted in Fig. 9 as dotted red curves for the 14 pump-probe distances experimentally studied. Bulk elastic parameters have been assumed for the tungsten and for the silicon. It is important to emphasize that the thickness of the tungsten film has been chosen as the thickness e = 252 nm experimentally deduced from the time-of-flight of the high frequency acoustic echoes. The magnitudes of both experimental and simulated results have been normalized. Having in mind that the same set of elastic parameters has been used in the calculations for every pump-probe distance, the agreement between experimental and calculated results is excellent. In particular, the celerity of both the SSLW and the PRW as well as the strong dispersion of the latter are perfectly reproduced by the simulations and are in agreement with results previously reported [42].
To emphasize the 48 MHz-spectral resolution of the asynchronous dual-oscillator system, the group velocity dispersion of the PRW has been analyzed in the spectral domain. For each pumpprobe distance z i (i = 1..14) considered in Fig. 9, the group velocity of the PRW is the frequency, has been extracted from the spectral phase φ i ( f ) of the Fourier transform of the experimental temporal response : v g,i ( f )=2πz i /(dφ i /df). The dispersion of v g,i is intrinsic to the geometry of the sample and does not depend on z i . Figure 10(a) shows thus the frequency dependence of the group velocity dispersion averaged over the 14 pump-probe distances. A monotonic decrease of the group velocity with increasing frequency is observed. Nevertheless, the dispersion relation of the group velocity of a supported thin film is non-monotonic with respect to the frequency [46,47]. We do not experimentally observe this non-monotonic evolution since the frequency content of the detected PRW only extends up to 1.2 GHz: It is imposed by the lateral size (≈ 1.5µm) of the focus pump spot on the free surface of the film, as explained in section 3.2. To confirm this particular behavior, we have thus calculated the derivative of the normal displacement for a lateral spot size of the opto-acoustic source reduced down to 200 nm in order to broaden the spectral content of the PRW up to 5 GHz. We have extracted the group velocity dispersion by using the formula given above. Figure 10(b) shows the evolution of the experimental group velocity (red dots) already plotted in Fig. 10(a) as well as of the calculated (blue open circles) group velocity with respect to the frequency. A minimum in the calculated group velocity is clearly visible close to 1.2 GHz and the limit at high frequencies tends towards the Rayleigh velocity in a tungsten half-space [dashed line in Fig. 10(b)]. Furthermore, experimental and calculated dispersions are in strong agreement concerning this spectral behavior as well as on the values of the velocities. Concerning the low frequency part of the spectrum, the calculated group velocity at 0.3 GHz is close to 3000 m.s −1 , i.e. well below the Rayleigh wave velocity at the free surface of a silicon half-space (≈ 5100 m.s −1 ). Lower frequencies must be detected to reveal the silicon anisotropy. This observation is in agreement with the almost perfectly circular acoustic wavefronts in Fig. 7. It is important to emphasize that the identification of the acoustic group velocity dispersion in the thin film is only possible by using low repetition rates lasers (48 MHz) to get a very high spectral resolution.

Conclusion
In conclusion, we have presented an ASOPS-based picosecond ultrasonics setup with 48 MHz repetition rates. The relative timing jitter has been quantified and is of the order of or lower than 500 fs over the whole pump-probe delays. This asynchronous dual-oscillator system offers thus at the same time a picosecond temporal resolution and a 48 MHz spectral resolution. The acoustic response of a supported thin film excited by an opto-acoustic point source has been investigated. Owing to the high temporal and high spectral resolutions of the ASOPS system high frequency acoustic echoes (up to 150 GHz) as well as low GHz SSLW and PRW could be imaged and group velocity dispersion of the PRW could be measured. A broadband optoacoustic imaging is therefore performed over more than two decades. The determination of the thin film thickness from the time-of-flight of the acoustic echoes combined with semi-analytical calculations perfectly reproduce the velocity of both SAWs and the strong dispersion of the PRW. The 48 MHz spectral sampling has permitted to experimentally extract the group velocity dispersion of the PRW from 0.3 to 1.2 GHz. Experimentally deduced and calculated dispersion are in strong agreement. Due to its picosecond temporal resolution, the ASOPS pump-probe setup presented here should allow to perform broadband opto-acoustic imaging up to 500 GHz with a 48 MHz spectral resolution.