Picosecond opto-acoustic interferometry and polarimetry in high-index GaAs

By means of a metal opto-acoustic transducer we generate quasi-longitudinal and quasi-transverse picosecond strain pulses in a (311)-GaAs substrate and monitor their propagation by picosecond acoustic interferometry. By probing at the sample side opposite to the transducer the signals related to the compressive and shear strain pulses can be separated in time. In addition to conventional monitoring of the reflected probe light intensity we monitor also the polarization rotation of the optical probe beam. This polarimetric technique results in improved sensitivity of detection and provides comprehensive information about the elasto-optical anisotropy. The experimental observations are in a good agreement with a theoretical analysis.


Introduction
In 1984 Thomsen et al. [1,2] developed the method of generation and detection of ultrashort coherent acoustic pulses in solids by femtosecond laser excitation. Thereby the energy of a short optical pulse being absorbed in the near-surface layer of a solid medium is converted into coherent lattice vibrations. The resulting elastic excitation is a strain pulse of picosecond duration with amplitude up to 10 -3 and ~10÷100-nm spatial extension, propagating through the crystal with sound velocity [2]. The excitation spot diameter is typically much larger than the penetration depth of light in the medium and thus the propagating pulse may be considered as a superposition of plane acoustic waves with wave vectors normal to the excited surface. This coherent wavepacket has a broad acoustic spectrum with frequencies up to hundreds of gigahertz (GHz). The composition by such high frequencies represents the main advantage of the femtosecond optical excitation compared to frequency-limited conventional techniques using piezoelectric transducers.
The detection of picosecond strain pulses is based on time-resolved monitoring of the strainmodulated reflectivity [2]. In an opaque material this detection occurs in the near-surface region providing information about the strain pulse amplitude and temporal shape. Shortly after development this method was extended by the technique of picosecond acoustic interferometry, which is suitable also for transparent media [3][4][5]. Here the intensity of coherent light reflected at an acoustic wavepacket propagating in the optically transparent medium oscillates in time with the frequency given by where  is the sound velocity, n is the refractive index of the medium,  is the wavelength of the reflected light in vacuum and is the incident angle of light inside the medium. The origin of the oscillations is dynamical interference of light partly reflected at the crystal surface with light reflected at the propagating strain pulse. These oscillations are commonly called Brillouin oscillations because f B is equal to the frequency shift in corresponding Brillouin scattering spectra for specular reflected light. Monitoring the Brillouin oscillations with picosecond time resolution allows one to obtain comprehensive information about the elastic, optical, electronic and mechanical properties of solid-state materials and structures [6][7][8][9][10][11][12][13][14][15], liquids [16][17][18] and biological objects [19,20].
In both cases, when the medium is either strongly absorbing or transparent for the probe light, its interaction with a coherent acoustic wavepacket is result of the elasto-optical effect, i.e. the strain-induced changes of the permittivity. The elasto-optical effect in general causes a medium to be optically anisotropic [21]. Thus, the changes of the intensity of the reflected light induced by the picosecond strain pulse become dependent on the polarization of the probe beam. Only for high-symmetry conditions, representing quite some restrictions, the strain-induced modulation of reflectivity remains insensitive to the probe light polarization: this is the case, for instance, for normally incident light scattered on a longitudinal acoustic wavepacket propagating along a high-symmetry crystallographic direction. Any reduction of symmetry, however, will make the optical probing polarization-sensitive.
The anisotropy of the elasto-optical interaction becomes especially important in experiments with picosecond shear strain pulses. The reflectivity signal induced by a transverse acoustic wavepacket depends on the polarization of the probe light. This effect has been addressed theoretically [21][22][23][24][25] and examined experimentally [23][24][25][26]. Experiments on near-surface detection of shear strain pulses have demonstrated also the possibility to extend the picosecond acoustic techniques by the methods of transient optical polarimetry, i.e. time-resolved monitoring of the ellipticity of the reflected, originally linearly-polarized probe light or the rotation of its polarization plane [27]. This possibility has been also theoretically analyzed [28].
Generation of shear strain pulses with picosecond duration, which has remained a challenge so far, may significantly extend the applications of high-frequency acoustics due to the smaller velocities and wavelengths of transverse acoustic waves. In related experiments [23][24][25][26][27] shear and compressive acoustic modes were generated simultaneously. They have different sound velocities, , resulting in different frequencies in the Brillouin signal, see Eq. (1) [23,25,26]. As generation and probing were performed in these experiments at the same spot of the studied sample one could not separate the contribution of compressive and shear acoustic modes in time, resulting in complex temporal signals. Thus, the shear component remained weak on the dominant background of the compressive contribution and therefore arduous for detection. This severely limits applications of picosecond acoustic interferometry using shear acoustic pulses.
Moreover, in all picosecond acoustic interferometry experiments, where Brillouin oscillations were measured, only intensity changes were monitored, while polarimetric techniques were not implemented so far.
The goal of the present work is to extend picosecond acoustic interferometry by polarimetric techniques and distinguish clearly the optical probe signals resulting from compressive and shear strain pulses by using a remote geometry. In this geometry generation and probing are clearly separated in space. In our experiments we generate quasi-longitudinal (QLA) and quasitransverse (QTA) strain pulses in a high-index GaAs substrate using a metal optoelastic transducer. The semiconductor substrate and the associated optical probe wavelength allow us to attain a large penetration depth for the probe light, thus, opening the possibility to detect the propagating acoustic wavepacket far away from the sample surface. Thus, we consider a material transparent for the probe light and focus only on the picosecond interferomic signal possessing Brilluoin oscillations with frequency given by Eq. (1). By comparing probe signals measured with traditional picosecond acoustic interferometry to signals obtained by polarimetric we show that the latter technique shows a much higher sensitivity to picosecond shear strain pulses and strongly depends on the orientation of the probe light polarization relative to the crystallographic axes of the sample. providing an excitation energy density of W=4 mJ/cm 2 . The strain pulse generation in the acoustically isotropic metal film by femtosecond optical excitation is well studied in literature [2,[29][30][31][32]. The ultrafast thermal expansion of the film leads to injection of  Comparing the panels (a) and (b) of Fig. 2 one sees that the amplitude of I QLA (t) for the QLA mode is about five times larger than the amplitude of I QTA (t) for QTA. The signals I QLA (t) and

Experiment
I QTA (t) are independent on  0 .

Theoretical analysis and discussion
The generation of the strain pulse takes place in the Al film deposited on the GaAs substrate. The thickness of the film (100 nm) is much larger than the penetration depth of the pump light (~10 nm), and assuming subsonic thermal diffusion we may consider two processes  independently: generation of the initial strain pulse in the Al film and then injection of this pulse into GaAs [25,31].
The hot carriers, optically generated in the narrow near surface region of aluminum (~10 nm), diffuse into the film and pass their energy to the lattice. The combined effect of the increased lattice temperature and the nonequilibrium carriers initiates the mechanical stress generating the strain pulse [2,[29][30][31][32]. The Al film is polycrystalline and behaves as an elastically isotropic medium, thus the initially generated strain pulse propagating normally to the surface is purely longitudinal (LA). From previous studies it is known, that such a pulse is bipolar and with good accuracy can be approximated by the derivative of a Gaussian function [36]. Figure 1b shows the calculated initial strain pulse  0 (t-z/ LA ), where  LA =6.3 km/s is the longitudinal sound velocity in Al [37].
The initial step of strain pulse generation in the Al film is followed by transmission of the "seed" strain pulse through the interface between the Al film and the acoustically anisotropic (311) GaAs substrate. In the coordinate frame shown in Fig. 1a The functions  p (t-z/ p ) describe the spatio-temporal evolution of the strain pulses and are determined by the "seed" pulse profile and its transmission through the Al/GaAs interface. The acoustic mismatch between aluminum and GaAs leads to partial reflection of the initial pulse at the interface. This reflection together with the acoustic mode conversion leads to appearance of a transverse pulse in the Al film ( TA =3.1 km/s in Al [37]) that is polarized along the x-axis in addition to the pure LA pulse. These two pulses propagate in the Al film back towards the surface with their individual sound velocities and after reflection there back towards GaAs. Then the transmission/reflection process accompanied by mode conversion repeats again. This complicated process of multiple injections may be addressed quantitatively in the frequency domain using standard elasticity theory, as described in Appendix A. The QLA and QTA strain pulses injected into the substrate perturb the dielectric permittivity of GaAs due to the photo-elastic effect [38]. The strain modifies the permittivity according to where  0 is the unperturbed dielectric permittivity, which is isotropic in GaAs, and p ijkl are the components of the photo-elastic tensor, written in the [311]-coordinate system. Since in our case there are only two nonzero strain components u xz (t,z) and u zz (t,z) for the geometry used in the experiments only two components,  xx (t,z) and  yy (t,z), have nonzero values. In the low-symmetry case these deviations are not equal, and the medium becomes optically anisotropic. Considering the spatio-temporal evolution of the strain we have to take into account that the strain pulses are reflected with -phase shift at the free surface of the GaAs substrate. Then using the Maxwell equations in linear to strain approximation we come to the expressions for the relative strain-induced perturbation of the probe pulse intensity and polarization plane rotation (see Appendix B): Here k=nk 0 +i and k 0 =2/ are the complex wave number of light in the substrate with refractive index n and in vacuum, respectively,  2 is the absorption coefficient, R EM and T EM are the complex reflection and transmission coefficients equal to T EM =2k 0 (k+k 0 ) -1 and R EM =(k 0 -k)(k+k 0 ) -1 . In the numerical calculations we use the value =2.210 6 m -1 [39].
The values of the photoelastic constants for GaAs are known [40] and allow us to get direct expressions for the dielectric permittivity perturbations: ). ,   The dependence of I p (t)/I 0 on  0 originates from the anisotropic polarization-depended term in Eq. (6) and is much weaker than for  p (t). It is expected that at  0 =/2 or 3/2 we should detect the maximal intensity modulation, while at  0 =0 or  the detected signals is minimal.
However, the maximal relative amplitude changes of I p (t)/I 0 measured at different angles are 5% for the QLA pulse and 20% for the QTA pulse. Expressed in absolute values, they are less than 210 -5 I 0 and arduous for detection [see the signals and their amplitudes in Figs. 2 (a) and 2(b)]. Thus, the angular dependence of I p (t) is not resolved in the experiment.

Conclusions
To summarize our experimental observations and theoretical analysis the optical anisotropy induced by the strain pulse propagating in GaAs along a high-index direction results in Brillouin oscillations of the polarization plane rotation of the probe light reflected at the propagating pulse.
In the particular low-symmetry case examined in our work this effect is strong and results in a higher sensitivity for polarimetric probing than in conventional picosecond acoustic interferometry where the modulation of the probe intensity is monitored. We operate with strain pulses of mixed polarization containing both compressive and shear strain components which are well separated in time due to the remote geometry in the experiments.
We want to point out also our analysis of the strain pulse generation. We have calculated the spatio-temporal profiles of the QLA and QTA strain pulses for a generation scheme with an optoelastic transducer on the high-index semiconductor substrate and estimated the strain pulse amplitudes and durations. The reasonability of our analysis is confirmed by the perfect quantitative agreement between calculated and measured signals, which are determined by the strain pulses parameters. The proposed scheme allows utilizing shear strain pulses for ultrafast manipulation of various excitations in nanostructures and thin films, since they may be easily integrated with a semiconductor substrate. An example is the recent magneto-acoustic experiments with shear strain pulses in ferromagnetic (311)-(Ga,Mn)As layer [43].
The hot carriers, optically generated in the narrow near surface region of the Al film, diffuse into the film and pass their energy to the lattice. The combined effect of the increased lattice temperature and nonequilibrium carriers initiates the mechanical stress generating the strain pulse. The Al film structure is polycrystalline, and we can treat it as an elastically isotropic medium. In this case the generated strain pulse is pure LA. In our case, however, the TA where  is necessary to consider self-consistently the complex processes of the nonequilibrium electron and lattice dynamics after ultra-fast laser pulse illumination. This problem is beyond the scope of this paper. Instead, we use the fact that the generated strain pulse has bipolar shape and can be approximated with good accuracy by the derivative of a Gaussian function [36]: where e is the base of the natural logarithm,  is the characteristic strain pulse duration and max zz u is the maximal value of the only strain component u zz along the propagation direction. This determines the spectral form of the "seed" pulse as , and the pulse duration  remains the only fitting parameter in our consideration. In the calculations we take =6.5 ps. Figure 1(b) shows the spatio-temporal shape of the initial pulse calculated with these parameters.
The second factor is determined by the interference of the LA and TA waves within the film and accounts for the multiple reflections and mode conversions at the interface. From Eq. (3a) we see that there are two series of resonances for which the emission into the substrate is facilitated. The frequencies of these "longitudinal" and "transverse" resonances are f LA(TA) =(1+2m) LA(TA) /(4d 0 ), where m is integer. The lowest resonances are f LA ≈15 GHz, 45 GHz and f TA ≈7.5 GHz, 22 GHz. In our case the transmission coefficients for the processes with mode conversion are smaller than those without conversion, and the same applies for the reflection coefficients. As a result, the QLA pulse is determined mainly by the first term in Eq.
(3a), and only "longitudinal" resonances are expected to be essential [see the inset in Fig. 1(c)].
For the QTA pulse, "transverse" resonances are also important, since both the first and the second terms contain reflection or transmission coefficients for the processes with mode conversion [see the inset in Fig. 1 (d)]. The spatio-temporal shapes of the strain pulses shown in Figs. 1 (c) and 1 (d) are obtained by the inverse Fourier transform of the corresponding spectra.

Appendix B. Interaction of light with the strain pulses
The strain pulses injected into the (311) GaAs substrate perturb the initially isotropic dielectric permittivity of the substrate through the photoelastic effect and as a result modulate the intensity and polarization of the reflected probe light pulse. We assume that strain pulses propagate through the substrate and reach its free surface without change of their shape. The characteristic time of strain variation is much greater than the duration of the probe light pulse. Therefore, calculating the characteristics of the reflected light at a particular pump-probe delay, we may assume the strain (and, consequently, also the dielectric permittivity) to be static and equal to its value for this time delay. For the temporal Fourier components of the electric field inside the substrate we have