1. Introduction

Picosecond acoustics is an opto-acousto-optical technique of characterization of mechanical properties of thin films [1,2]. The thickness of opaque thin films in multilayers can be measured by this method, which is based on the optical detection of picosecond longitudinal acoustic echoes from the interfaces. The acoustic pulses are generated by pump laser beam. The echoes are generally detected by measuring the transient reflectivity of the sample through the femtosecond pump-probe technique [3]. Transient interferometry techniques [4–8] can also be applied to detect the normal displacement that occurs when an acoustic pulse reaches the free surface of the sample. Deflection [9] and beam distortion techniques [10] can alternatively be used to detect both longitudinal and shear picosecond acoustic pulses.

The efficient generation and detection of picosecond shear acoustic pulses is a more difficult task than the generation and detection of picosecond longitudinal acoustic pulses. So far, only few experiments involving shear waves in the GHz-THz range have been reported [11–17]. Advances in generation and detection of shear waves at high frequencies will be beneficial to the investigations of condensed matter. Ultrashort shear acoustic pulses could then be applied for the study of rheological properties of liquids and polymers at high frequencies.

Several theoretical approaches were developed to model the ultrafast modulation of the reflected optical field induced by picosecond shear acoustic pulses that propagate within a sample [13–15]. The modulation of the reflected electric field is generally expressed in terms of the variation Δr or the relative variation Δr/r = Δρ/ρ + iΔθ of the reflection coefficient r = ρe^iθ. The experimental results obtained with transient reflectometry are generally expressed as the relative variations of the reflectance ΔR/R = 2 Δρ/ρ whereas in transient interferometry the transient phase signal Δθ is measured. In the case of an optically anisotropic sample, the representation of the ultrafast optical response by one reflection coefficient r for each possible state of linear polarization, i.e. p or s, e is no longer sufficient because of the coupling that occurs between polarization states during reflection. For a probe beam at oblique incidence, there are two eigenstates of polarization: in the p-polarized optical mode the electric field lies in the plane of incidence whereas the electric field is perpendicular to the plane of incidence for the s-polarized mode. Thus, in addition to the reflection coefficients r_pp and r_ss for p- and s-polarized probe, the optical properties of the sample must be described by the scattering coefficients r_sp and r_ps which characterize the amplitude of the optical fields which are scattered from p to s polarization and vice versa. The four coefficients r_pp, r_ss, r_sp and r_ps are the components of the reflection Jones matrix of the sample. The advantage of the reflection matrix formalism is to handle at once the polarization effects via the scattering coefficients r_sp and r_ps as well as the variations in amplitude of the reflected field represented by the coefficients r_pp and r_ss.

In this paper, we propose a theory of detection of picosecond shear acoustic pulses which describes the ultrafast optical response with a Jones matrix. There is evidence that shear waves may couple the p and s polarization modes. Hence, the scattering coefficients r_sp and r_ps may be non zero in the presence of shear waves. Indeed, acousto-optic modulators or deflectors that involve shear acoustic waves are able to switch the polarization from p at the input to s polarization at the output, and vice versa [18,19].

The presented theory will be restricted to samples characterized by an isotropic permittivity tensor. In the following, such samples will be called optically isotropic. The adverb “optically” is important, as the sample may anisotropic for another physical property. This is in particular the case for cubic crystals which are optically isotropic but may have an anisotropic elastic tensor. Moreover, we will restrict the study to samples which do not scatter light, i.e. perfectly reflecting sample, so the Jones matrix formalism is sufficient to properly deal with the problem [20]. The case of diffusing samples would require the more general Mueller matrix formalism. Section 2 will introduce the transient reflection matrix which is a normalized Jones reflection matrix to describe the transient optical properties of the sample. The relationship of its off-diagonal components with the complex rotation of polarization will be pointed out. Then, the relevance of transient polarimetry techniques to detect shear waves in optically isotropic samples will clearly appear. In section 3, the transient reflection matrix will be calculated in the case of a cubic crystal to predict the polarimetric signals that are expected to be measured. Then the optimal experimental conditions to probe shear acoustic waves will be presented.

2. The transient reflection matrix

In picosecond acoustics experiments, we consider an acoustic plane wave that is generated at the surface z=0 of a semi-infinite (z>0) absorbing medium by a short optical pump pulse (Fig. 1 ). A second optical pulse after the pump pulse is used to probe the acoustic waves, which can be detected within the absorption depth l_a=λ/(4πn”) of the substrate, where $\text{n}={\text{n}}^{\prime }–\text{i}{\text{n}}^{″}$ (n”>0) is the refractive index of the absorbing medium for the probe wavelength λ, and vacuum wave vector k=2π/λ. The medium (z<0) covering the opaque medium (z>0) is transparent with a refractive index n₀. The plane of incidence of the optical probe beam is the ZX-plane. The incidence angle is ${\varphi }_0$, whereas the transmitted beam is refracted at the angle ϕ (Fig. 1).

 

Fig. 1 Pump-probe configuration used in picosecond acoustics. A probe beam at oblique incidence is used to detect an ultrashort acoustic pulse located at a distance z below the interface z=0.

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We suppose that the problem is one-dimensional. However, a description in 3 dimensions would be essential if the pump and probe laser beams are both sharply focused on the sample surface [15]. The incident and reflected probe fields are represented by the Jones vectors E _i and E _r respectively. The relationship E _r=R⋅E _i defines the reflection matrix R of the sample. The Jones vectors and the R Jones matrix are explicitly expressed in terms of the p and s polarization components as follows:

In the subsequent sections, we will consider that the unperturbed sample is optically isotropic, so r_ps = r_sp = 0 and therefore the TRM matrix simplifies as follows:

The normalized Jones vector J can be expressed in terms of the two angular parameters ψ and χ which define the polarization ellipse (Fig. 2 ) [20]:

 

Fig. 2 The polarization ellipse. V₁ and V₂-axes coincide respectively with the p and s-polarizations. The V₁’ and V₂’ axes are the principle axes of the ellipse. The angular parameters ψ and χ are: (ψ,χ)_p=(0,0) for p polarization and (ψ,χ)_s=(π/2,0) for s polarization.

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The diagonal components represent the relative change of the complex amplitude of the reflected field: (Δr/r)_p,s =(ΔA/A)_p,s =(ΔA₀/A₀+iΔα)_p,s. The off-diagonal components: (ΔΩ)_p,s= (Δψ + iΔχ)_{p, s} represent the complex rotation of polarization. The comparison of Eqs. (3) and (6) leads to:

In magneto-optic studies, the complex rotation of polarization is either a Kerr or a Faraday rotation of polarization [21,22,]. In electro-optic sampling studies, the transient electric fields in semiconductors can be sampled by measuring the transient birefringence, which is in most cases equivalent to the measurement of ellipticity changes induced by the electro-optic effect [23]. Another example is the monitoring of the rotation relaxation of molecules in a liquid by measuring the transient birefringence of the sample [24].

The knowledge of the transient reflection matrix allows one to express the transient optical response for any arbitrary state of polarization of the incident probe light. For example, the detection configurations denoted 45-p or p-u in the experiments of Refs [14,25]. can be analyzed with the Jones formalism. In [14,25], the 45-p configuration means that incident probe is linearly polarized at 45° from the plane of incidence - (ψ,χ)=(45°,0) - and the reflected probe is detected through an analyzer which transmits only the p-component E_rp of the reflected field. Thus the Jones vector for the incident field polarized at 45° can be written in the form E _i = (1,1)^T, where the superscript T denotes the transposition of the row vector (1,1). The reflected field is E _r =(r_pp, r_ss)^T, if r_ps=r_sp=0, according to Eq. (1). Indeed the last assumption is almost valid for the SiO₂/Zn sample of [25] because of the intrinsic weak optical birefringence of the zinc crystal. Thus the transient relative intensity on the photodetector in the 45-p configuration is: (ΔI/I)_p=2 Re(ΔE_rp/E_rp)=2 (ΔA₀/A₀)_p + 2 Δψ_p. The configuration p-u (p-polarized incident probe and no analyzer) of Ref [14,25]. leads to the transient relative intensity (ΔI/I)_u=2 (ΔA₀/A₀)_p. The shear pulses were detected in the transparent SiO₂ film with the 45-p configuration but not with the p-u configuration. Thus, the shear wave signal is included in the rotation of polarization Δψ_p only, whereas the longitudinal wave signal is contained in the term 2 (ΔA₀/A₀)_p. The Jones matrix formalism clearly shows that the shear wave that propagates in the SiO₂ film induces a rotation of polarization of the reflected probe. Hence, the opportunity to detect selectively shear waves would be provided by a direct measurement of the rotation of polarization, i.e. by transient polarimetry.

The measurement of the rotation of polarization [16] was applied to shear acoustic echoes that are detected almost at the same time as longitudinal echoes [17]. Several experimental setups can be used to measure the rotation of polarization Δψ or the ellipticity changes Δχ; they are presented in Refs [16,21–24]. In the next section, we will demonstrate that the measurement of the complex rotation of polarization is a relevant method to detect shear acoustic waves.

3. Shear wave detection

To calculate the Δ_R matrix of the sample perturbed by an arbitrary strain profile, we start by calculating the response to a delta-strain pulse.

If the transparent medium covering the opaque medium is an isotropic liquid or an amorphous solid, shear waves may propagate in this medium and may be detected. In general, the acoustic attenuation at frequencies above 10 GHz is so great that the ultrashort pulses cannot propagate at distances longer than a few micrometers [26]. In the following, we will restrict the study to the detection of acoustic waves in the opaque medium (z>0).

We assume that the opaque medium is an optically isotropic solid. This assumption implies that the solid medium may be either an amorphous solid or a cubic crystal. In general, the elastic waves that propagate in a cubic crystal are not always purely transverse or longitudinal, i.e. they are quasi-transverse (QT) or quasi-longitudinal (QL) [27]. The displacement vectors of the QT and QL waves have both transverse and longitudinal components. However, when the waves propagate in a direction parallel to a symmetry axis of the crystal, the QT and QL waves degenerate as pure transverse and longitudinal waves. In the following, we will assume that the Z-axis coincides with the [001] crystal axis of the cubic crystal. This assumption implies that the acoustic velocity of shear waves is independent of the direction of the transverse displacement vector u _T in the XY-plane. This velocity is $c_T=\sqrt{C_{44}/\rho }$, where C₄₄ is a component of the elastic tensor and ρ is the density of the medium.

We must also consider the symmetry properties of the photoelastic tensor p_ijkl which governs the optical detection. This tensor is defined by the relationship: $\Delta {\eta }_{ij}=p_{ijkl}{S}_{kl}$, where $\Delta {\eta }_{ij}$ and $S_{kl}$ are respectively the perturbation of the impermeability tensor and the strain tensor. The Voigt notation will be used to express explicitly the tensor relationship. If the medium is amorphous, the photoelastic tensor is determined by the two independent components p₁₁ and p₁₂ [28]. For a cubic crystal which belongs to one of the following crystal classes: 432, m3m, and 4̄3m, the three required independent components are p₁₁, p₁₂ and p₄₄. We suppose that the [001] crystal axis coincides with the Z-axis whereas the [100] crystal axis forms an angle γ with the X-axis. The strain tensor of a pure transverse wave that propagates along the Z-axis is defined by the two components: S₄ =∂u_Y/∂z and S₅=∂u_X/∂z, where u_X = u_T cos φ and u_Y = u_T sin φ are the components of the transverse displacement vector u _T in the XY-plane. The angle φ characterizes the direction of the displacement vector u _T relative to the plane of incidence. Using formulas for the rotation of the photoelastic tensor [28], one can express the tensor relationship: $\Delta {\eta }_{ij}=p_{ijkl}{S}_{kl}$, in the XYZ Cartesian frame as follows:

Let us calculate the perturbation ΔR of the reflection matrix due to a delta strain pulse located within the substrate at a depth z (Fig. 1). To explain the mechanism of detection of a delta strain perturbation, we follow the same reasoning as in Ref [3]. The transmission of the incident field at the z=0 interface towards the medium z>0 is expressed by the transmission matrix T ₀. After propagation over a distance z, characterized by the propagation matrix P ₀, the optical wave interacts with the δ-strain pulse. The discontinuity of the permittivity tensor at the z-plane induces a partial reflection characterized by the δR _T reflection matrix. Then, the reflected field propagates towards the z=0 interface, characterized by the propagation matrix P ₁. Finally, the transmission of the field into the transparent medium n₀ is characterized by the transmission matrix T ₁. Therefore the ΔR perturbation matrix is expressed as a product of Jones matrices as follows:

The δR _T reflection matrix has to be evaluated at the δ-discontinuity. We consider a shear strain of magnitude S_T(z)=∂u_T/∂z, where u_T is the magnitude of the transverse displacement in the XY-plane. The magnitude of the delta-like S_T(z) strain at z is such that ${\displaystyle {\int }_0^{\infty }{\text{S}}_{\text{T}}(z)dz}={\text{u}}_{\text{T}}$, so u_T represents here the magnitude of the step-like shear displacement localized at a depth z within the sample. The δR _T matrix can be calculated using a 4 x 4 matrix formalism [20,29,30]. The strain perturbation is first considered as a birefringent slab of thickness e and then the δR _T matrix is calculated for the limit e→0. The result of the calculation, exact to the first order in u_T, is:

To calculate explicitly (Δ_R)_Tδ, i.e. the matrix (Δ_R)_T for a delta-like shear strain pulse, we need the expression of the diagonal matrices: T ₀=[t_0p, t_0s], T ₁=[t_1p, t_1s], and the unperturbed diagonal reflection matrix R=[r_p, r_s], where:

t_0p= 2n₀ cosϕ ₀ / (n₀ cosϕ + n cosϕ ₀), t_0s= 2n₀ cosϕ ₀ / (n₀ cosϕ ₀ + n cosϕ), t_1p= 2n cosϕ / (n₀ cosϕ + n cosϕ ₀), t_1s= 2n cosϕ / (n₀ cosϕ ₀ + n cosϕ). r_p= (n₀ cosϕ - n cosϕ ₀) / (n₀ cosϕ + n cosϕ ₀), r_s= (n₀ cosϕ ₀ - n cosϕ) / (n₀ cosϕ ₀ + n cosϕ).Equation (10) leads to:

The important result here is that all the diagonal components of the (Δ_R)_T matrix are zero. This result means that the shear waves have no effect on the amplitude of the reflected field. Therefore shear plane waves cannot be detected by transient reflectometry or interferometry techniques, independently of the choice of the probe incidence angle and the probe polarization. In consequence, the only way to detect a shear plane wave -not a QT wave - that propagates normally to the surface in an optically isotropic medium is to use a detection configuration which is sensitive to the rotation of polarization of the reflected probe. On the contrary, the detection of QT waves is possible with transient reflectometry or interferometry, with a probe beam at normal incidence [13]. Moreover, the detection of a shear wave, which reaches at oblique incidence the surface of an isotropic medium, can be achieved by transient interferometry, with an optical probe at normal incidence [15]. So transient polarimetry would not provide a significant advantage for the detection of QT wave or plane shear waves at oblique incidence. In contrast, polarimetry is useful only if the component of the displacement vector of the considered wave is zero in the direction of the surface normal.

The expression of the a_T coefficient, which determines the detection sensitivity, shows that a shear wave is not detected if the transverse displacement u _T is parallel to the plane of incidence, i.e. for φ=0°, but the sensitivity reaches a maximum for φ=90°, i.e. when the displacement vector u _T is perpendicular to the plane of incidence. In addition, the sensitivity vanishes for normal incidence (ϕ ₀=0° and ϕ=0°), it is thus essential to probe shear waves at oblique incidence.

For simulations, it is convenient to set: n cos ϕ = (n² - n₀ ² sin² ϕ ₀)^1/2 = ν’ - iν”, as n₀ sinϕ ₀ = n sinϕ, with in order to ensure the damping of the optical field in the opaque medium for z>0.

The off-diagonal components of the matrix $h_{T0}\cdot e^{-2ikn\cos \phi z}$ in Eq. (12) or Eq. (13) determine the complex rotation of polarization per unit of transverse displacement, which is expressed as:

As an example, we calculated the polarimetric signals Δψ and Δχ per unit of u_T, for both p and s polarizations, induced by a shear delta strain pulse within a Gallium Arsenide (GaAs) substrate with (100) orientation in air (n₀=1). The Δψ and Δχ polarimetric signals of Eq. (14) are represented in Fig. 3 . The transverse displacement is supposed to be perpendicular to the plane of incidence. The refractive index of GaAs at the probe wavelength 800 nm (E=1.552 eV) is n=3.684-0.092i [32]. The photoelastic coefficient p₄₄ = −0.46-0.21i for GaAs at 800 nm is calculated from the piezo-optical constant π₄₄=(1.5+0.5i).10⁻⁹ Pa⁻¹ of Ref [33]. and the elastic constant C₄₄= 59.44 GPa [32] by the relationship: p₄₄ = – (π₄₄ C₄₄) / n⁴. Equation (14) shows that each polarimetric signal oscillates in function of z with the spatial pseudo-period Λ = λ /(2ν′) . The oscillations are damped with a characteristic length L = λ / (4πν″), which depends both on the refractive index of the material and the incidence angle. For either p or s polarizations, the Δψ and Δχ oscillations have the same magnitude H but are phase-shifted by π/2. As a shear pulse propagates with a velocity c_T, the polarimetric signals oscillate at the Brillouin frequency: f_B = c_T / Λ = (2c_T ν′) / λ, and the oscillations decay with the characteristic damping time τ = L / c_T. When a shear pulse is reflected at the free surface z=0, a change of sign occurs in the displacement vector u _T. Consequently, the signs of Δψ and Δχ are changed after reflection. This results in a step in the polarimetric signals recorded as a function of time. Figure 3 shows that, at z = 0, Δχ signal has a greater magnitude than the Δψ signal. Hence, the step of the shear pulse echo at the free surface would be greater with the Δχ signal. depends on If the phase θ _p,s, is zero or 180°, then the amplitude of the step Δψ signal would be zero and the echo would not be detected. On the contrary, the Δχ signal, phase shifted by π/2, would lead to a step of maximum amplitude. It is therefore more advantageous to measure both the complementary signals: Δψ and Δχ.

 

Fig. 3 Rotation of polarization for p and s polarization. The transverse displacement vector of the δ-like shear strain pulse is perpendicular to the plane of incidence (φ=90°).

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Figure 4(a) represents the magnitude H of the Brillouin oscillations as a function of the incidence angle ϕ ₀. For a p-polarized probe, a resonance occurs at the Brewster angle (ϕ₀≈74.8°). Nevertheless, probing the shear waves with p polarization will be impossible around the Brewster angle, because of the exceedingly small reflectance of the substrate (Fig. 4b). In conclusion, it would be preferable to probe the shear pulse with s polarization at an incidence angle around ϕ₀≈56° for which the magnitude of the Brillouin oscillations reaches a maximum. At the optimum angle of incidence ϕ₀≈56°, the Brillouin frequency and the decay time, are respectively 30.0 GHz and 202 ps, since a shear wave velocity along a <100> crystallographic axis of GaAs is c_T = 3343.5 m/s.

 

Fig. 4 (a) Magnitude of the Brillouin oscillations, for p and s probes at 800nm.(b) Reflectance curves of GaAs at 800 nm. The symbol r denotes the complex reflection coefficient either for p or for s polarizations.

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We will consider now the possibility of measuring polarimetric signals induced by longitudinal waves. The δR _L reflection matrix for a longitudinal delta-strain pulse, corresponding to a step-like displacement u_L, located within the substrate at depth z, is calculated. Here, the only non zero component of the strain tensor is S₃ =∂u_Z/∂z. Hence, Eq. (9) shows that the photoelastic constants involved in the detection process are p₁₁ and p₁₂. The calculation leads to:

After replacing the δR _T matrix in Eq. (10) by δR _L,, one can calculate the (Δ_R)_L matrix for the longitudinal strain pulse. As all the matrices in Eq. (10) are diagonal, the (Δ_R)_L matrix is also diagonal. Consequently, the longitudinal waves do not induce any polarimetric signal, as the off-diagonal components of the (Δ_R)_L matrix are zero. This property is of practical importance if an echo of a shear pulse has to be detected with polarimetry at the same time as an echo of a longitudinal pulse. The diagonal components of the (Δ_R)_L matrix calculated by Eqs. (10) and (13), provided that the subscript T is replaced by L, is in agreement with Eqs. (15) and (16) of [34], assuming that the normal displacement of the z=0 surface is zero.

We consider now the effect of thermal transients on polarimetric signals. The generation of acoustic pulses by an optical pump pulse absorbed at the surface of the sample generates inevitably a quick rise of the sample temperature that is followed by a temperature decay due to the diffusion of heat within the substrate. Since the refractive index is temperature-dependent, the thermal signal is always present in transient reflectivity signals so the acoustic signals appear frequently superposed to the thermal signal. As we consider that the heated medium is thermally and optically isotropic, the (Δ_R)_θ matrix, where the subscript θ denotes the temperature, must be diagonal in the case of homogeneous heating along the XY-plane. Hence, thermal transients can be in principle cancelled in polarimetric measurements.

4. Conclusion

The Jones formalism has been used as a tool to demonstrate the close relationship that exists between shear strain and the induced perturbations of the polarization states of a probe beam. The measurement of the complex rotation of polarization is quite relevant for the detection of shear waves whereas the longitudinal waves have no effects on the rotation of polarization. The detection of a plane shear wave with transient polarimetry techniques would be useful for the studies of the rheological properties of liquids and soft matter. Moreover, the transient polarimetry could be applied to enhance the detection sensitivity of some particular vibration modes of nanoparticles, quantum dots, or in general nanostructures that involve shear waves. For instance, the torsional modes of a spherical particle could in principle be detected using this technique. The combination of transient polarimetry with other optical technique would be profitable for future research in picosecond acoustics but also in other fields of research.