A specific feature of SAM, when strongly focused transducers are utilized for imaging, is that the imaging contrast varies considerably with the distance between the acoustic lens and the specimen, enabling the visualization of subsurface regions depending on the acoustic properties of the specimen
25 and the numerical aperture of the acoustic lens. Acoustic waves incident at the Rayleigh critical angle, excite Rayleigh surface acoustic waves (RSAW), provided that the surface of the imaged sample supports the propagation of such waves. Rayleigh waves can be excited and detected by appropriate defocusing utilizing an acoustic lens with a sufficiently large aperture opening angle that exceeds the critical angle for exciting Rayleigh waves.
Figure 1 illustrates the excitation and detection of RSAW by defocusing the acoustic lens toward the sample.
26 Ray 1, which is incident on the sample at the Rayleigh angle excites a RSAW which propagates in the surface of the sample and leaks energy back into the couplant and thus, excites a compressional wave in the fluid under the Rayleigh angle. A ray that propagates along a path symmetrical to ray 1, (ray 4) to the transducer interferes with the specular ray (ray 3). The voltage value of the resulting (summed) signal, therefore depends on the lens-sample distance (
z) and the wavelengths in the couplant. As the spacing “z” changes, the relative phases of the rays 1 and 4 vary, so that the superposition alternates between constructive and destructive interference. This effect considerably contributes to the contrast in the acoustic micrographs. A plot over the defocus results in an interference pattern referred to as the
V(
z) curve containing minima and maxima according to the interference of the two wave modes.
27,28 The period of the resulting oscillations Δ
z, in the
V(
z) curve, can be calculated from the phase relations and forms the base for calculating the velocity of the RSAW
26 in the sample surface. The
V(z) signature can finally be presented mathematically as
29:
\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\unicode[Times]{x1D6C2}}\)\(\def\bupbeta{\unicode[Times]{x1D6C3}}\)\(\def\bupgamma{\unicode[Times]{x1D6C4}}\)\(\def\bupdelta{\unicode[Times]{x1D6C5}}\)\(\def\bupepsilon{\unicode[Times]{x1D6C6}}\)\(\def\bupvarepsilon{\unicode[Times]{x1D6DC}}\)\(\def\bupzeta{\unicode[Times]{x1D6C7}}\)\(\def\bupeta{\unicode[Times]{x1D6C8}}\)\(\def\buptheta{\unicode[Times]{x1D6C9}}\)\(\def\bupiota{\unicode[Times]{x1D6CA}}\)\(\def\bupkappa{\unicode[Times]{x1D6CB}}\)\(\def\buplambda{\unicode[Times]{x1D6CC}}\)\(\def\bupmu{\unicode[Times]{x1D6CD}}\)\(\def\bupnu{\unicode[Times]{x1D6CE}}\)\(\def\bupxi{\unicode[Times]{x1D6CF}}\)\(\def\bupomicron{\unicode[Times]{x1D6D0}}\)\(\def\buppi{\unicode[Times]{x1D6D1}}\)\(\def\buprho{\unicode[Times]{x1D6D2}}\)\(\def\bupsigma{\unicode[Times]{x1D6D4}}\)\(\def\buptau{\unicode[Times]{x1D6D5}}\)\(\def\bupupsilon{\unicode[Times]{x1D6D6}}\)\(\def\bupphi{\unicode[Times]{x1D6D7}}\)\(\def\bupchi{\unicode[Times]{x1D6D8}}\)\(\def\buppsy{\unicode[Times]{x1D6D9}}\)\(\def\bupomega{\unicode[Times]{x1D6DA}}\)\(\def\bupvartheta{\unicode[Times]{x1D6DD}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bUpsilon{\bf{\Upsilon}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\(\def\iGamma{\unicode[Times]{x1D6E4}}\)\(\def\iDelta{\unicode[Times]{x1D6E5}}\)\(\def\iTheta{\unicode[Times]{x1D6E9}}\)\(\def\iLambda{\unicode[Times]{x1D6EC}}\)\(\def\iXi{\unicode[Times]{x1D6EF}}\)\(\def\iPi{\unicode[Times]{x1D6F1}}\)\(\def\iSigma{\unicode[Times]{x1D6F4}}\)\(\def\iUpsilon{\unicode[Times]{x1D6F6}}\)\(\def\iPhi{\unicode[Times]{x1D6F7}}\)\(\def\iPsi{\unicode[Times]{x1D6F9}}\)\(\def\iOmega{\unicode[Times]{x1D6FA}}\)\(\def\biGamma{\unicode[Times]{x1D71E}}\)\(\def\biDelta{\unicode[Times]{x1D71F}}\)\(\def\biTheta{\unicode[Times]{x1D723}}\)\(\def\biLambda{\unicode[Times]{x1D726}}\)\(\def\biXi{\unicode[Times]{x1D729}}\)\(\def\biPi{\unicode[Times]{x1D72B}}\)\(\def\biSigma{\unicode[Times]{x1D72E}}\)\(\def\biUpsilon{\unicode[Times]{x1D730}}\)\(\def\biPhi{\unicode[Times]{x1D731}}\)\(\def\biPsi{\unicode[Times]{x1D733}}\)\(\def\biOmega{\unicode[Times]{x1D734}}\)\begin{equation}\tag{1}V\left( z \right) = 2\pi \mathop \int \nolimits_0^{\pi \over 2} P\left( \theta \right)R\left( \theta \right)exp\left[ { - i2kz\cos \theta } \right]\cos \theta \sin \theta d\theta \end{equation}
where
P(
θ) is the pupil function that expresses the angular emission and detection properties of the acoustic lens, and
θ as the angle of incidence. The integral in
Equation 1 considers the overall geometrical contribution that ranges from 0° to the opening of the semiaperture
θSA,
R(
θ) is the reflectance function, describing the amplitude and the phase of reflected waves as a function of the incidence angle
θ, and (
k) is the propagation factor, defined as
k = 2
πf/
v , with
f being the acoustic frequency and
V0 the velocity of propagation of the acoustic wave in the coupling fluid. The periodicity Δ
z of the oscillations in the
V(
z) curve is defined as:
\begin{equation}\tag{2}\Delta z = {{\lambda _0} \over {2\left( {1 - cos{\theta _R}} \right)}}\end{equation}
where,
λ0 is the wavelength of the acoustic waves during propagation in the coupling fluid and
θR is the Rayleigh critical angle. The velocity of propagation of the Rayleigh surface acoustic waves
VR can be expressed in terms of the ratio to the critical angle
θR as:
\begin{equation}\tag{3}{V_R} = {\raise0.7ex\hbox{${V_0}$} \!\mathord{\left/ {\vphantom {{V_0}\, {\sin {\theta _R}}}}\right.\kern-1.2pt}\!\lower0.7ex\hbox{${\,\sin {\theta _R}}$}}\end{equation}
The velocity of Rayleigh surface acoustic waves is related to the velocity of the shear (
Vs) and compressional waves (
Vl) propagating in isotropic samples, for the same Possion's ratio as
29,30:
\begin{equation}\tag{4}{\left( {{{V_R} \over {V_s}}} \right)^6} - 8{\left( {{{V_R} \over {V_s}}} \right)^4} + 8\left[ {3 - 2{{\left( {{{V_s} \over {V_l}}} \right)}^2}} \right]{\left( {{{V_R} \over {V_s}}} \right)^2} - 16\left[ {1 - {{\left( {{{V_s} \over {V_l}}} \right)}^2}} \right] = 0\end{equation}