Dynamic reflectivity of an opto-acoustic three-layer transducer for picosecond ultrasonic measurements

Le Ridant, Louise; Ponge, Marie-Fraise; Audoin, Bertrand

doi:10.1063/5.0184672

We present an analysis of a three-layered structure used for both launching and detecting picosecond acoustic waves. To enhance the optical sensitivity to the acoustic disturbances, a cavity is designed, which acts as a Fabry–Perot interferometer. We use analytic modeling based on dual optical and acoustic transfer matrix formalism to analyze the coupled optical and acoustic wave propagation. Assuming a three-layer transducer made of an optical cavity sandwiched between two thin metallic layers, the model allows for mastering of the coupled optical and acoustic responses, which leads to an optimum design of the structure, and it highlights the various acoustic contributions to the reflectivity changes. The sensitivity of this three-layered structure to acoustic disturbances is compared to the numerical predictions we performed for the standard opto-acoustic transducer made of a single metallic layer.

I. INTRODUCTION

Picosecond ultrasonics is an optical technique that can be used for the remote measurement of the mechanical properties of thin layers. It is based on a pump-probe setup using femtosecond laser pulses. Acoustic waves in the GHz frequency range are generated by the absorption of ultra-short laser pulses. Acoustic perturbations can be detected through their interaction with another laser beam. The relative reflectivity change in the structure can then be measured.^1–3 The sub-micrometric spatial resolution and the non-invasive nature of the technique enable the mechanical characterization of micrometric structures. Picosecond ultrasonics has been used to investigate the thermal and acoustic properties of materials⁴ and to determine the thickness, the speed of acoustic waves,⁵ and the acoustic attenuation⁶ of opaque, semi-transparent, and transparent layers.^7,8 This technique allows for the characterization of many sorts of materials: from thin metallic and crystalline layers^9–13 to polymers¹⁴ and amorphous solids.⁷ Recently, the mechanical properties of biological cells^15–20 have been studied. Understanding how cells adhere to substrates and knowing their stiffness and adhesion strength are crucial in order to comprehend the fundamental biological processes.²¹ By measuring the reflection coefficient of the generated stress pulse at the cell/substrate interface, it is now possible to map the acoustic impedance of a dehydrated cell placed on a transducer composed of a metallic layer sputtered onto a transparent substrate.¹⁶ To distinguish the cell clearly, the difference in acoustic impedance, and thus in mechanical properties (density and elastic moduli), between the cell and its environment has to be sufficiently high. In the case of a hydrated cell in an aqueous environment, this difference is very small. Moreover, the detection of small changes in acoustic reflectivity due to minor variations in the mechanical parameters of the prospected cell requires an enhancement of the signal to noise ratio. To do this, it has been previously shown that it is possible to take advantage of the high sensitivity of an optical cavity to variations in its thickness.²² Optical cavity resonance takes place in a layer whose thickness is equal to or greater than half of the optical wavelength and smaller than the optical penetration depth. The intensity of the reflected and transmitted beams depends on the thickness of the cavity with respect to the optical wavelength, the reflectivity of the interfaces, and the optical absorption. Enhancement of the detection of the acoustic signal, thanks to the optical cavity, has already been achieved in an optical fiber hydrophone by in vivo ultrasound dosimetry during clinical extracorporeal shock-wave lithotripsy²³ and in a photoacoustic scanner for fast 3D imaging of the soft tissue structure.²⁴ The sensitivity of an air cavity has been analyzed to increase the amplitude of signals measured in aluminum and copper films.²⁵ In addition, a three-layer transducer has been optimized and used for reflection measurements to investigate phonon propagation in water²⁶ and for transmission measurements to detect Brillouin scattering in biological cells.²⁷ The design of the optical cavities was based on optical and acoustical considerations to maximize the quality factor of the optical cavity and to define suitable cavity spacing. It relied on calculations of the static reflectivity and transmissivity of the cavity. However, reflectivity changes in the structures caused by the optically induced acoustic transient waves were not predicted. In this work, we therefore present a simulation intended to predict the dynamic reflectivity change in a three-layered structure and to quantify the gain one can expect compared to a usual monolayer transducer.

In this article, we first present a general optical and acoustic modeling, both based on transfer matrix formalism, to predict the dynamics of optical and acoustic waves propagating through a multi-layered structure. We then describe the modeling of the optical detection of acoustic transient waves through a three-layer transducer. Finally, we investigate the calculated waveforms resulting from the dynamic reflectivity change in the three-layered structure and quantify the resulting gain compared with a monolayer transducer.

II. OPTO-ACOUSTIC GENERATION IN A MULTI-LAYERED STRUCTURE

To understand the physical mechanisms that underlie wave propagation in a layered opto-acoustic transducer, let us briefly consider a single metallic layer with free boundary conditions. The optical energy of a sub-picosecond pump laser pulse is absorbed in the layer and is converted into heat. If thermal diffusivity is ignored, the temperature increases within the limit of the absorption zone. A thermal stress is created, and, consequently, a strain pulse is generated. Neglecting spreading of the pulses as they pass through the optical components, the temporal length τ of the incident optical beam is assumed to be less than one picosecond. For metallic materials, the product of the acoustic velocity by this duration v_Lτ is less than 6 nm, and the optical absorption depth is around 10–30 nm. Therefore, the frequency bandwidth of the generated acoustic waves depends on the optical absorption depth and is above 10 GHz. The acoustic nano-pulse propagates back and forth through the film. The lateral dimension of the focused laser spot on the surface is larger than micron; thus, it is much larger than the acoustic wavelength, the value of which is typically twice the optical absorption in metallic materials.² Consequently, we can assume that only plane longitudinal waves are generated and that the thermoelastic generation process can be modeled using a one-dimensional model.

We now present this theoretical model of the thermoelastic generation process in a multi-layered structure. This structure is shown in Fig. 1, where the black arrows → (the “head” of this arrow must be like the ones that are associated with a_i and b_i in Fig. 1), and → represent the electric and acoustic fields that propagate through the structure, respectively, which consists of N layers and is sputtered onto a transparent substrate denoted medium 0. A sample, layer N + 1, is placed on the stack. The spatial axis $\overset{̆}{z}$ lies along the stacking direction of the multi-layer. The origin $\overset{̆}{z} = 0$ is at the interface between the multi-layer and the substrate. The incident light comes from $\overset{̆}{z} < 0$ with normal incidence and propagates through the multi-layer in the positive $\overset{̆}{z}$ direction. z is the local spatial axis of each layer. z_i ∈ [0, d_i], with $d_{i} = {\overset{̆}{z}}_{i} - {\overset{̆}{z}}_{i - 1}$ being the thickness of layer i (i ∈ [1, N]). Layers 1 to N − 1 are optically transparent or semi-transparent, unlike the N^th layer, which is opaque. Thus, the incident light propagates through the N − 1st layers and is entirely absorbed by the N^th layer. Both media 0 and N + 1 are half-spaces of the electric and acoustic fields.

FIG. 1.

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The multi-layered structure composed of N layers sputtered onto a substrate, medium 0. A sample, denoted N + 1, is placed on the stack. The normal optical pump beam is represented by the black arrows → (the head of this arrow must be like the ones that are associated with a_i and b_i in the figure), and the generated acoustic waves are represented by the black arrows →. The thickness of each layer i (i ∈ [1, N]) is defined by $d_{i} = {\overset{̆}{z}}_{i} - {\overset{̆}{z}}_{i - 1}$ ⁠, with ${\overset{̆}{z}}_{i}$ and ${\overset{̆}{z}}_{i - 1}$ being the positions of the interfaces of layer i along the spatial global axis $\overset{̆}{z}$ of the entire multi-layer. z is the local spatial axis for each layer, where z_i ∈ [0, d_i] is shown as an example for layer N.

In the following, we first present the calculation of the optical field in the whole structure using the transfer matrix formalism.^9,28 Then, based on the density of electromagnetic energy absorbed in each layer, we describe the temperature field in the multi-layered structure. Finally, we present the acoustic model also based on transfer matrix formalism to calculate the acoustic perturbation propagating in the layered structure.

A. Electromagnetic waves in the layers

The pump and probe laser beams propagate in the whole structure up to the N^th layer. In layer i (i ∈ [0, N]), the electric field is expressed as

E_{i} (z) = a_{i} e^{- j q_{i} z_{i}} + b_{i} e^{j q_{i} z_{i}}

⁠, where q_i = n_iq_v is the optical wavenumber in layer i, with n_i being the refractive index and q_v = 2π/λ_v being the optical wavenumber in vacuum. λ_v stands for the wavelength of the optical wave in vacuum. a_i and b_i are the amplitudes of the forward and backward propagating optical waves, respectively (Fig. 1). As the tangential components of the electric and magnetic fields are continuous at the interface

{\overset{̆}{z}}_{i - 1}

⁠, the amplitudes of the electric field in the ith layer can then be calculated from the amplitudes in the (i − 1)th layer. By iterating this reasoning, it is possible to connect the amplitudes of the electric field in the (i + 1)th layer to the amplitudes of the electric field in medium 0,²⁹

[\begin{matrix} a_{i + 1} \\ b_{i + 1} \end{matrix}] = M^{0, i + 1} [\begin{matrix} a_{0} \\ b_{0} \end{matrix}],

(1)

with

M^{0, i + 1} = \prod_{j = 1}^{i} P_{j, j + 1} V_{j - 1, j + 1} P_{0, 1}

(2)

being the transfer matrix. The matrix

V_{j - 1, j + 1}

expresses the propagation of the field through layer j. It depends on the optical wavenumber q_j and on the layer thickness d_j.

P_{j, j + 1}

accounts for the interaction of the optical field with the interface between layers j and j + 1. It depends on the optical reflection coefficient r_j,j−1 = (n_j − n_j−1)/(n_j + n_j−1) and on the transmission coefficient t_j,j−1 = 2n_j/(n_j + n_j−1).

For a known amplitude a₀ of the incident electromagnetic wave, application of Eq. (1) requires the reflection and transmission coefficients to be calculated for the whole structure.² For this, the amplitudes of the electric field in medium 0 are connected to those in layer N with the transfer matrix M^N,0. Since b_N = 0, the optical reflection coefficient of the multi-layered structure is $r_{0,N}^{L} = b_{0} / a_{0} = M_{21}^{N, 0} / M_{11}^{N, 0}$ ⁠, and the optical transmission coefficient of the structure is $t_{0,N}^{L} = a_{N} / a_{0} = 1 / M_{11}^{N, 0}$ ⁠, with $M_{l, m}^{N, 0}$ (l = 1, 2; m = 1, 2) being the components of the transfer matrix M^N,0. It is then possible to predict the electric field inside the structure and to obtain the optical reflection and transmission coefficients of each layer.

B. Temperature increase in the layers

The density of energy is related to the Poynting vector by the local equation of conservation of energy. In the considered multi-layer, the thickness of the absorbing layers is smaller than the optical absorption depth. The reflection of the pump beam on the back interface of these absorbing layers must therefore be taken into account in the calculation. Thus, the density of electromagnetic energy in a semi-transparent layer i is expressed as

\begin{array}{lc} Q_{i} (z, t) & = \frac{n_{i, p}^{'} v_{0} ε_{0} β_{i, p}}{2} [| a_{i} |^{2} e^{- β_{i, p} z} + | b_{i} |^{2} e^{β_{i, p} z} \\ + 2 R (a_{i} b_{i}^{*}) \cos (2 q_{i, p}^{'} z) + 2 I (a_{i} b_{i}^{*}) \sin (2 q_{i, p}^{'} z)] f_{t} (t), \end{array}

(3)

where

n_{i, p}^{'}

is the real part of the optical refractive index at the wavelength of the pump beam, v₀ is the celerity of the optical wave in vacuum, ɛ₀ is the vacuum dielectric constant, and

β_{i, p} = 2 q_{i, p}^{″}

is the absorption coefficient for the pump beam.

q_{i, p}^{'}

and

q_{i, p}^{″}

are the real and imaginary parts of the optical wavenumber of the pump beam in the i^th layer, respectively.³⁰ f_t(t) is the time profile of the light pulse and is equal to the Dirac distribution δ(t).

The first two terms in Eq. (3) account for the absorption of the electric field in the i^th layer. The other terms correspond to stationary waves. They represent the distribution of energy as discrete and spatially periodic sources buried deep into the layers. The density of energy depends on the amplitude of the forward and backward components a_i and b_i of the electric field in the i^th layer. Since layer N is opaque, the electric field has no backward components (b_N = 0) and the density of energy absorbed in the layer is represented by the first term of Eq. (3) only, which is the expression given in the literature for an opaque layer.¹

In this work, as thermal diffusivity has a minor impact on the waveform of the acoustic wave, we consider that the temperature increase takes place only in the absorbing layers, i.e., the semi-transparent and opaque layers. The temperature increase is therefore related to the density of the electromagnetic field by the heat equation without diffusivity. It is then proportional to the density of the electromagnetic energy and is calculated by integrating the density of energy according to time,^1,2

T_{i} (z, t) = \frac{1}{ρ_{i} C_{p, i}} Q_{i} (z) H (t),

(4)

where ρ_i is the density, C_p,i is the specific heat capacity, and H(t) is the temporal Heaviside step function.

Finally, if layer i is in the stack (1 ≤ i ≤ N), the optical amplitudes a_i and b_i depend strongly on the reflection and transmission coefficients on both of its sides due to the left and right parts of the stack since $a_{i} = t_{0,i}^{L} a_{0}$ and $b_{i} = r_{i,N}^{L} a_{i}$ ⁠, where $t_{0,i}^{L}$ and $r_{i,N}^{L}$ are the optical transmission coefficient of layers from substrate 0 to layer i and the reflection coefficient from layer i to layer N, respectively. The temperature field can therefore be predicted and controlled in the absorbing layers by adjusting the thickness of each layer.

C. Acoustic waves in the multi-layered structure

To calculate displacement in the ith layer, we use the equation of propagation of acoustic waves with the spatial gradient of temperature as a source term.² The equation is solved in the Fourier domain, and the general solution is

{\tilde{u}}_{i} (z, ω) = {\tilde{U}}_{i}^{h -} e^{- j k_{i} z} + {\tilde{U}}_{i}^{h +} e^{j k_{i} z} + S_{p, i}^{D} (z, ω),

(5)

with

{\tilde{U}}_{i}^{h -} e^{- j k_{i} z} + {\tilde{U}}_{i}^{h +} e^{j k_{i} z}

being the complementary function and

\begin{matrix} S_{p, i}^{D} (z, ω) = & \tilde{H} (ω) [{\tilde{U}}_{i}^{β -} e^{- β_{i, p} z} + {\tilde{U}}_{i}^{β +} e^{β_{i, p} z} \\ + {\tilde{U}}_{i}^{c} \cos (2 q_{i, p}^{'} z) + {\tilde{U}}_{i}^{s} \sin (2 q_{i, p}^{'} z)] \end{matrix}

(6)

being the particular integral.

{\tilde{U}}_{i}^{h -}

and

{\tilde{U}}_{i}^{h +}

are the amplitudes of the forward and backward propagating acoustic waves of the complementary function, respectively, and k_i is the acoustic wavenumber in the i^th layer.

{\tilde{U}}_{i}^{β -}

⁠,

{\tilde{U}}_{i}^{β +}

⁠,

{\tilde{U}}_{i}^{c}

⁠, and

{\tilde{U}}_{i}^{s}

are the amplitudes of the particular integral of the displacement. They are calculated by the insertion of Eq. into the propagation equation. Calculating of the amplitudes

{\tilde{U}}_{i}^{h -}

and

{\tilde{U}}_{i}^{h +}

of the complementary function involves the continuity of the displacements and stresses at the interfaces,

{\tilde{u}}_{i - 1} ({\overset{̆}{z}}_{i - 1}) = {\tilde{u}}_{i} ({\overset{̆}{z}}_{i - 1})

and

{\tilde{σ}}_{i - 1} ({\overset{̆}{z}}_{i - 1}) = {\tilde{σ}}_{i} ({\overset{̆}{z}}_{i - 1})

⁠. In contrast to the optical wave, where the incident wave comes from medium 0, generation of the incident elastic wave occurs in layer 1 because the substrate (medium 0) is transparent. Consequently, the amplitudes of the complementary function in layer i + 1 can be obtained from the amplitudes in layer 1 and are expressed with the formalism by Sun et al.,³¹

[\begin{matrix} {\tilde{U}}_{i + 1}^{h -} \\ {\tilde{U}}_{i + 1}^{h +} \end{matrix}] = M_{1, i + 1}^{D} [\begin{matrix} {\tilde{U}}_{1}^{h -} \\ {\tilde{U}}_{1}^{h +} \end{matrix}] + Ω_{1, i + 1}^{D},

(7)

with

M_{1, i + 1}^{D} = \prod_{j = 1}^{i} P_{j, j + 1}^{D} V_{j - 1, j + 1}^{D}

(8)

being the acoustic transfer matrix. The propagation matrix

V_{j - 1, j + 1}^{D}

expresses the propagation of the acoustic wave across the jth layer. It is dependent on the acoustic wave number k_j+1 and the thickness of layer j + 1, d_j+1. The interface matrix

P_{j, j + 1}^{D}

expresses the interaction of the acoustic field with the interface between layers j and j + 1. It is dependent on the acoustic reflection

r_{j + 1, j}^{D} = (Z_{j + 1} - Z_{j}) / (Z_{j + 1} + Z_{j})

and transmission

t_{j + 1, j}^{D} = (2 Z_{j + 1}) / (Z_{j + 1} + Z_{j})

coefficients, with Z_j and Z_j+1 being the acoustic impedance of layers j and j + 1, respectively.

The term

Ω_{1, i + 1}^{D}

in Eq. comes from the particular integral and is linked to the thermal source term. It is expressed as

\begin{align} Ω_{1, i + 1}^{D} & = \sum_{j = 1}^{i} \{\prod_{m = j}^{i - 1} P_{m + 1, m + 2}^{D} V_{m - 1, m + 1}^{D} (Π_{j - 1, j} ({\overset{̆}{z}}_{j - 1}) + ϒ_{j - 1, j} ({\overset{̆}{z}}_{j - 1}))\} \\ + {Π_{i, i + 1} ({\overset{̆}{z}}_{i}) + ϒ_{i, i + 1} ({\overset{̆}{z}}_{i})} . \end{align}

(9)

It is dependent on terms $Π_{i, i + 1} ({\overset{̆}{z}}_{i})$ and $ϒ_{i, i + 1} ({\overset{̆}{z}}_{i})$ ⁠. $Π_{i, i + 1} ({\overset{̆}{z}}_{i})$ is composed of the particular integral $S_{p}^{D}$ and its derivative according to space $S_{p}^{D^{'}}$ ⁠. $ϒ_{i, i + 1} ({\overset{̆}{z}}_{i})$ represents the thermo-elastic strains. It is dependent on the amplitude of temperature $\tilde{T}$ at the ${\overset{̆}{z}}_{i}$ interface. Thermal strain is the sum of both these terms at each encountered interface. They also contribute to the calculation of the amplitudes of the complementary function at the considered interface, here at ${\overset{̆}{z}}_{i}$ ⁠.

For the optical model, amplitudes are calculated at both extremity interfaces (z = z₀ and z = z_N). There is no incident wave coming from medium 0, so

{\tilde{U}}_{0}^{h -} = 0

⁠, and there is no left-propagating wave in medium N + 1, so

{\tilde{U}}_{N + 1}^{h +} = 0

⁠. We have

[\begin{matrix} {\tilde{U}}_{N + 1}^{h -} \\ 0 \end{matrix}] = M_{0, N + 1}^{D} [\begin{matrix} 0 \\ {\tilde{U}}_{0}^{h +} \end{matrix}] + Ω_{0, N + 1}^{D},

(10)

with

M_{0, N + 1}^{D}

being the acoustic transfer matrix that takes into account propagation through both 0/1 and N/N + 1 interfaces and

Ω_{0, N + 1}^{D}

being the thermo-elastic vector. With these calculations, the partial acoustic waves into each layer of the structure can be predicted.

III. ELASTO-OPTIC DETECTION IN A THREE-LAYER TRANSDUCER

We now consider the propagation of acoustic waves in a three-layered structure composed of a transparent layer (layer 2) sandwiched between two metallic layers (layers 1 and 3), as shown in Fig. 2. The stack is sputtered onto a substrate (medium 0), considered a transparent half-space. The pump and probe beams go through the stack from medium 0 to the left interface of layer 3. The physical quantities associated with the pump and probe beams are labeled with subscripts “p” and “s”, respectively. The thickness of layer 1 is smaller than the optical penetration depth to let in both laser beams, and layer 3 is thicker than the optical penetration depth. Thus, the pump laser is absorbed in layer 1 and in layer 3. The absorbed energy densities in these layers are dependent on the stacking reflectivity and transmittivity, respectively. The latter are strongly related to the layer thicknesses. Displacement of the layer’s surfaces induced by the transient temperature increases, and subsequent thickness changes, has no influence on the pump energy absorptions because the increase in the time of the transient thermal strain is larger than the pump pulse duration.

FIG. 2.

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Three-layered structure on a substrate (medium 0). It is composed of two metallic layers (layers 1 and 3) and a transparent layer (layer 2) of respective thickness d₁, d₃, and d₂. A pump beam (red lines) and a probe beam (green lines) penetrate through the structure up to layer 3. The position of the acoustic strain is indicated by coordinate z′.

The probe beam propagates through the transducer up to opaque layer 3, with neglected absorption. It then probes along the whole stack and into layer 3 within the limit of its optical absorption depth.

Without acoustic perturbations, the probe beam is reflected on the three-layered structure with the static optical reflection coefficient $r_{0, 3}^{L}$ of the set of the two layers {layer 1, layer 2}. With acoustic perturbations, the dynamic reflectivity of the structure reveals two kinds of acousto-optic interactions. The first is the reflectometric contribution. It is related to the local variation in the refractive index Δn caused by the acoustic waves in the layers. The optical wave interacts with the local optical change caused by the stress pulse within the limit of the absorption zone of each layer. The elasto-optic interaction is related to the acoustic strain by a sensitivity function and is weighted by the elasto-optic coefficient that quantifies the sensitivity of the refractive index to the acoustic strain.¹ The sensitivity function is specific to the considered structure and connects the reflectivity variations to the perturbations of the refractive index. The second acousto-optic interaction is the interferometric contribution: the reflectivity variation caused by the displacement of each interface. For each interaction, we consider the reflectivity change Δr caused by the presence of the acoustic disturbance, which is the difference between the dynamic reflectivity and the static reflectivity. The formulation of this interaction at the surface of an absorbent layer was first carried out by Thomsen in 1986.¹ Other calculations were performed to describe the interaction with the acoustical pulse in a transparent half-space³² and a transparent layer.³³ Furthermore, the interaction in a multi-layer structure was studied analytically by Perrin et al.³⁴ and by Matsuda and Wright et al. with Green’s functions.³⁰

In Subsection III A, the interaction of the optical beam with the acoustic perturbation in each layer is described. We first present the modeling of the interactions between the optical wave and the acoustic waves in the transparent substrate (medium 0) and in the semi-transparent and transparent layers (layers 1 and 2). We then explain the calculation of the reflectivity change in the opaque layer (layer 3) and the reflectivity variations caused by the interferometric interaction. Finally, we analyze the total reflectivity of the three-layer transducer.

A. Elasto-optic interaction with the acoustic strain inside each layer of the three-layer transducer

1. Reflectivity change of a transparent substrate and a transparent layer

In transparent media, the probe beam interacts with the optical perturbation along the entire length of the layer. The change in the refractive index is modeled using a virtual interface located at coordinate z′, as shown with vertical dotted lines in Fig. 2.

Since the substrate is considered as a half-space, mechanical strain propagates in the substrate from the interface substrate/layer 1 toward the depth of the substrate. At the first order of interaction with the local optical perturbation, three kinds of interactions, shown in Fig. 3, are distinguished, as described in Refs. 32 and 35. The resulting change in reflectivity is written as follows:

\begin{matrix} Δ {\tilde{r}}_{0} (ω) & = & j q_{0, s} \frac{\partial n_{0, s}}{\partial η} \int_{- \infty}^{0} {\tilde{η}}_{0} (z^{'}, ω) [e^{2 j q_{0, s} z^{'}} + r_{0, 3}^{L 2} e^{- 2 j q_{0, s} z^{'}} + 2 r_{0, 3}^{L}] d z^{'}, \end{matrix}

(11)

where q_0,s,

\frac{\partial n_{0, s}}{\partial η}

⁠, and

{\tilde{η}}_{0} (z^{'}, ω)

are, respectively, the optical wavenumber of the probe beam, the elasto-optic coefficient, and the strain in the substrate. The first term of the integrand in Eq. represents the reflection on the front side of the optical perturbation shown with arrow (a) in Fig. 3. The optical wave is only reflected at the optical perturbation, unlike the reflection on the rear side of the optical perturbation and the partial transmission through the optical perturbation shown by arrows (b) and (c) in Fig. 3. They are both reflected from layer 1 and depend on the optical reflection coefficient

r_{0, 3}^{L}

of the bilayer {layer 1, layer 2}, as expressed by the second and third terms of the integrand in Eq. .

FIG. 3.

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Interaction of the optical probe beam with the virtual interface at z′ in a transparent half-space with layer 1 sputtered onto it: (a) forward interaction (solid lines); (b) backward interaction (dotted lines); (c) transmission through the perturbation (dashed line).

In this configuration, the transparent layer is delimited by two metallic layers on which the optical beam is multiply reflected. As the optical beam is confined between the two reflectors, the structure acts as an optical cavity. Because of the cavity resonance, the optical reflection and transmission coefficients of this layer are very sensitive to variation in its thickness. It is important to notice that the thickness of layer 1, through which the probe beam passes, is assumed to be smaller than the penetration depth in a metallic layer, which is itself considered smaller than the optical wavelength used, q₁d₁ ≪ 1. Thus, there is no cavity resonance in layer 1, but its thickness modifies its optical reflection and transmission coefficients and, by extension, $r_{0, 3}^{L}$ ⁠. The total reflectivity given by Eq. (11) would therefore take into account any optical reflection and transmission coefficient variations due to the variation in the thickness of layers 1 and 2.

The exponential factors of the first and second terms of the integrand represent the extension and reduction of the optical path compared to the optical path without the strain pulse. Both these terms depend on the strain pulse position. These waves interfere with the optical waves partially reflected upon the three-layered structure. The interferences are alternately constructive and deconstructive, and the reflected intensity oscillates with time. These fluctuations are called Brillouin oscillations, as they can also be understood in the context of Brillouin scattering.^2,35 The third term of the integrand represents the partial transmission through the optical perturbation. The interaction is the same at any position of the stress pulse, and the reflectivity change is proportional to the displacement of the interface $\int_{- \infty}^{0} {\tilde{η}}_{0} (z^{'}, ω) d z^{'} = u_{0} - u_{\infty}$ and u_∞ = 0.

For transparent layers with finite thicknesses, a second reflection must be considered.³³ In this case, the reflectivity changes in layer 1 and layer 2 are expressed by Eqs. and , respectively:

\begin{align} Δ {\tilde{r}}_{1} (ω) & = j D_{1} q_{0, s} \frac{\partial n_{1, s}}{\partial η} \int_{0}^{d_{1}} {\tilde{η}}_{1} (z^{'}, ω) [e^{- 2 j q_{1, s} z^{'}} \\ + (r_{1, 3}^{L 2} e^{- 2 j q_{1, s} (d_{1} - z^{'})} + 2 r_{1, 3}^{L}) e^{- 2 j q_{1, s} d_{1}}] d z^{'}, \end{align}

(12)

\begin{align} Δ {\tilde{r}}_{2} (ω) & = j D_{2} q_{0, s} \frac{\partial n_{2, s}}{\partial η} \int_{0}^{d_{2}} {\tilde{η}}_{2} (z^{'}, ω) [e^{- 2 j q_{2, s} z^{'}} \\ + (r_{2, 3}^{2} e^{- 2 j q_{2, s} (d_{2} - z^{'})} + 2 r_{2, 3}) e^{- 2 j q_{2, s} d_{2}}] d z^{'} . \end{align}

(13)

In Eqs. and , the first and second terms of the integrand describe the Brillouin interaction with the optical perturbation at z′. They express the reflection on the left and right sides of the optical perturbation, respectively. The last term is the partial transmission through the optical perturbation. The exponential factors represent the extension of the optical path compared to a simple reflection on the substrate/layer 1 interface for Eq. , and on the layer 1/layer 2 interface for Eq. . Factors D₁ and D₂ express the multiple reflections and transmissions at both sides of both layers, respectively. However, as indicated in the beginning of Sec. III, there is no cavity effect in layer 1. Only factor D₂ is related to the optical resonance depending on the thickness/wavelength ratio, via q_2,sd₂,

D_{2} = \frac{t_{0, 2}^{L} t_{2,0}^{L}}{{(1 - r_{2, 3} r_{2,0}^{L} e^{- 2 j q_{2, s} d_{2}})}^{2}} .

(14)

Equation also shows that the reflection and transmission coefficients of layer 1,

r_{0, 2}^{L}

and

t_{0, 2}^{L}

⁠, modify the reflectivity of layer 2. Thus, there is an optical coupling between the two layers, which acts on the reflectivity of the probe beam via the reflection and transmission coefficients of the layers.

2. Reflectivity change of the opaque layer

For an opaque medium such as layer 3, the optical beam penetrates into the layer within the limit of the optical penetration depth and interacts with the strain directly under the interface. In the situation where the optical penetration depth is less than half the optical wavelength, there are neither Brillouin oscillations nor optical cavity effect. The variation in the reflectivity is expressed by the following equation:²

Δ {\tilde{r}}_{3} (ω) = j q_{0, s} t_{0, 3}^{L} t_{3,0}^{L} \frac{\partial n_{3, s}}{\partial η} \int_{0}^{d_{3}} {\tilde{η}}_{3} (z^{'}, ω) e^{- 2 j q_{3, s} z^{'}} d z^{'} .

(15)

The term

e^{- 2 j q_{3} z^{'}}

expresses the extension of the optical path from the interface between layer 2 and layer 3 to the position of the stress pulse, as shown in Fig. 2. The reflectivity change is proportional to the elasto-optic coefficient ∂n_3,s/∂η, which also depends on the coupled optical effects occurring in the bilayer {layer 1, layer 2} due to the transmission coefficients

t_{0, 3}^{L}

and

t_{3,0}^{L}

⁠.

B. Reflectivity change due to interface motion

The optical waves are reflected on the interface between the substrate and layer 1, and the phase of the reflectivity depends on the displacement of this interface, u_0/1. For small phase shifts, the reflectivity change due to the moving interface is²

Δ r_{disp,0} (t) = - 2 j q_{0, s} r_{0, 3}^{L} u_{0 / 1} (t),

(16)

and it depends on the optical reflection coefficient of the bilayer {layer 1, layer 2},

r_{0, 3}^{L}

⁠.

The reflectivity of layers 1 and 2 is sensitive to the phase changes created by the dynamic variations in their thickness.³⁶ The reflectivity change resulting from the variation in thickness Δd_i of a transparent layer i delimited by two layers i − 1 and i + 1, i = 1, 2, is given by

Δ r_{disp,i} = \frac{\partial r_{i - 1,i + 1}^{L}}{\partial d_{i}} Δ d_{i},

(17)

with

r_{i - 1,i + 1}^{L}

being the optical reflection coefficient of layer i. Moreover, the variation in thickness is the difference in the displacements of the two interfaces on either side of layer i, u_i−1/i and u_i/i+1. To calculate the reflectivity change in layer 1, according to Eq. , we differentiate

r_{0, 2}^{L}

with respect to d₁. The variation in reflectivity Δr_disp,1 is expressed as

Δ r_{disp,1} = - j 2 q_{1, s} D_{3} (u_{1 / 2} - u_{0 / 1}),

(18)

with

D_{3} = \frac{t_{0, 1} t_{1, 0} r_{1, 3}^{L} e^{- 2 j q_{1, s} d_{1}}}{{(1 - r_{1, 0} r_{1, 3}^{L} e^{- 2 j q_{1, s} d_{1}})}^{2}} .

(19)

In Eq. , D₃ depends on the optical reflection coefficient

r_{1, 3}^{L}

and thus on the thickness of layer 2. For the same reasons, the reflectivity of layer 2 is

Δ r_{disp,2} = - j 2 q_{2, s} D_{4} (u_{2 / 3} - u_{1 / 2}),

(20)

with

D_{4} = \frac{t_{0, 2}^{L} t_{2,0}^{L} r_{2, 3} e^{- 2 j q_{2, s} d_{2}}}{{(1 - r_{2,0}^{L} r_{2, 3} e^{- 2 j q_{2, s} d_{2}})}^{2}} .

(21)

In Eq. , the reflectivity change Δr_disp,2 is proportional to D₄, which depends on the optical reflection and transmission coefficients

t_{02}^{L}

⁠,

t_{20}^{L}

⁠, and

r_{20}^{L}

⁠, and thus on the thickness of layer 1. Hence, the coupling effect between the two layers is also clearly visible in factors D₃ and D₄ via the optical reflection and transmission coefficient of each layer.

C. Dynamic reflectivity of the three-layer transducer

The total dynamic reflectivity Δr of the three-layer transducer is the sum of the dynamic reflectivity of each layer,

Δ r (t) = \sum_{i = 0}^{3} Δ r_{i} (t) + \sum_{i = 0}^{2} Δ r_{disp,i} (t) .

(22)

The reflectivity change due to the strain pulse in each layer is calculated in the Fourier domain and then in the time domain. For the dynamic reflectivity generated by the interface motion, displacements are calculated in the Fourier domain and then in the time domain, and the associated reflectivity change is directly calculated in the time domain.

During opto-acoustic measurements, the photodetector detects the dynamic variations in the intensity of the probe beam. The change in intensity of the reflected probe beam of the three-layer stack is $Δ R = ∣ r_{0, 3}^{L} + Δ r ∣^{2} - ∣ r_{0, 3}^{L} ∣^{2}$ ⁠. To be free from the influence of the intensity of the probe beam, the relative reflectivity change $Δ R / R_{0, 3}^{L}$ was then calculated, with $R_{0, 3}^{L} = | r_{0, 3}^{L} |^{2}$ being the static optical reflectance of the three-layer transducer for the probe beam.

In this section, we have seen that the reflectivity changes in each layer are the result of the two kinds of acousto-optic interactions and are sensitive to the variations in thickness of both layers. The combined effect of the thickness dependence, the optical cavity effect in the transparent layer 2, and the dynamic displacements of the interfaces due to the propagation of the acoustic waves modifies the reflection and transmission coefficients of each layer, and consequently the total reflectivity change. Furthermore, close to the optical resonance, the total reflectivity of the structure is very sensitive to the dynamic variation in the optical cavity thickness that takes place in layer 2. In the following, we intend to use this effect that has been previously used in other studies to strongly enhance the total reflectivity changes in samples and structures.²⁵

IV. SIMULATIONS AND RESULTS

The previous calculations were used to analyze variations in the reflectivity change of a three-layer transducer and to quantify the gain with respect to the reflectivity change predicted for a single-layer transducer. We were therefore interested in improving the detection of the acoustic pulses that propagate back and forth in layer 3 and arrive at the interface between layers 2 and 3. Both the amplitude of the acoustic waves and the sensitivity to relative reflectivity change must be maximized. For this, the pump has to be transmitted largely through the bilayer {layer 1 and layer 2}, in order to have the best generation possible in layer 3, and the sensitivity of the probe to acoustic disturbances must be maximum. To enhance the sensitivity of the reflectivity to the variation in thickness, we used the combined effect of the influence of the thickness of layer 1 and the optical cavity resonance in layer 2, pointed out in Eq. (20). At a given optical wavelength, the reflectivity of an optical cavity varies strongly with cavity thickness, especially at the resonance peak where its derivative with respect to thickness is maximum. Thus, to improve the detected signal, the thicknesses of layer 1 and layer 2 must be chosen so that they reach the strong variation point of the reflectivity of the whole structure while, at the same time, maximizing the pump transmission.

In this work, the optical wavelength in vacuum of the pump and probe beams was, respectively, λ_v,p = 1030 nm and λ_v,s = 515 nm. The three-layer transducer considered here was on a sapphire (Al₂O₃) half-space and was composed of one transparent layer of lithium fluoride (LiF) between two metallic layers of titanium (Ti). LiF was chosen for its low refractive index compared to titanium. This difference in the refractive index maximizes the reflection of light between the two metallic layers and the quality factor of the optical cavity. The optical and mechanical properties of each material at the given optical wavelengths are listed in Table I.

TABLE I.

Properties of the materials used for simulations. Parameters marked with ∗ are obtained by fitting the model to the experimental data. C′ and γ are, respectively, the real part of the elastic constant and the dynamic attenuation, where C = C′ + jωγ.³⁷ Averaged values were considered for anisotropic materials.

	Sapphire	Titanium	LiF
$n_{λ_{515}}^{'}$	1.75	1.78	1.39
$n_{λ_{1030}}^{'}$	1.75	2.95	1.37
$ξ_{λ_{515}}$ (nm)	0	17.1	0
$ξ_{λ_{1030}}$ (nm)	0	21.3	0
∂n/∂η	0.08*	10 + j9*	0.3*
κ (W m⁻¹ K⁻¹)	35	21.9	11.3
C_p (J kg⁻¹ K⁻¹)	761	528	1562
α (K⁻¹)	5.8 × 10⁻⁶	9 × 10^–6	37 × 10^–6
ρ (g cm⁻³)	3.98	4.5	2.63
C′ (GPa)	481	220*	112
γ (GPa s)	0	0.0221*	0

	Sapphire	Titanium	LiF
$n_{λ_{515}}^{'}$	1.75	1.78	1.39
$n_{λ_{1030}}^{'}$	1.75	2.95	1.37
$ξ_{λ_{515}}$ (nm)	0	17.1	0
$ξ_{λ_{1030}}$ (nm)	0	21.3	0
∂n/∂η	0.08*	10 + j9*	0.3*
κ (W m⁻¹ K⁻¹)	35	21.9	11.3
C_p (J kg⁻¹ K⁻¹)	761	528	1562
α (K⁻¹)	5.8 × 10⁻⁶	9 × 10^–6	37 × 10^–6
ρ (g cm⁻³)	3.98	4.5	2.63
C′ (GPa)	481	220*	112
γ (GPa s)	0	0.0221*	0

In this section, we present the optical reflection and transmission coefficients of the three-layer Ti–LiF–Ti on a sapphire half-space calculated with the optical model described in Subsection II A for the probe and pump beams, respectively. Next, we analyze the calculated relative reflectivity change. Finally, we quantify the predicted enhancement of detection sensitivity.

A. Optical static reflectivity and transmission of a three-layer transducer

In order to enhance the sensitivity of the reflectivity to dynamic variations in thickness, we began by analyzing the variations in the sensitivity of the relative reflectivity change of the Al₂O₃–Ti–LiF–Ti structure with layer 1 and 2 thicknesses. We then considered the sensitivity of the reflectance according to the thickness of layer 2, $\partial R_{0, 3}^{L} / \partial d_{2}$ ⁠. As said before, during opto-acoustic measurements, the photodetector detects the dynamic variations in the intensity of the probe beam; sensitivity was thus divided by the static reflectance of the structure $R_{0, 3}^{L}$ ⁠. This division improves the sensitivity of the structure: the lower the static reflectance, the higher the sensitivity.

Figure 4(a) presents variations in the sensitivity of the relative change in reflectivity, $1 / R_{0, 3}^{L} \times \partial R_{0, 3}^{L} / \partial d_{2}$ vs d₂ and d₁. We can see that the shape of the sensitivity according to d₂ is bipolar due to the rising and falling sides of the reflectance near the resonance. Moreover, d₁ modifies the optical reflection coefficient of layer 1 and, therefore, the reflectivity of the bilayer. Sensitivity extrema are obtained for d₁/λ_1,s = 0.041, where d₁ = 12 nm, and for several thicknesses of layer 2. The thickness d₁ of 12 nm is smaller than the optical penetration depth in titanium, i.e., d₁β₁ = 0.70 (Table I). Thus, as expected, the probe beam reaches the transparent LiF layer of the transducer. The first extremum occurs for a thickness d₂ close to λ_2,s/4, and each following pair of extrema is spaced by λ_2,s/2. In order to maximize the sensitivity of the reflectivity to the dynamic variation in d₂, the thicknesses of both layers must be adjusted.

FIG. 4.

(a) Sensitivity of the relative reflectivity change to d2, 1/R0,3L×∂R0,3L/∂d2, according to d1 and d2; (b) transmittance of the pump beam T0,3L (solid line) vs the ratio d2/λ2,p, and sensitivity of the relative reflectivity change in the probe beam to d2 (dotted line) vs d2/λ2,s for d1 = 12 nm. The red dots highlight the value of the transmittance of the pump beam for the positive extrema of sensitivity.

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(a) Sensitivity of the relative reflectivity change to d₂, $1 / R_{0, 3}^{L} \times \partial R_{0, 3}^{L} / \partial d_{2}$ ⁠, according to d₁ and d₂; (b) transmittance of the pump beam $T_{0, 3}^{L}$ (solid line) vs the ratio d₂/λ_2,p, and sensitivity of the relative reflectivity change in the probe beam to d₂ (dotted line) vs d₂/λ_2,s for d₁ = 12 nm. The red dots highlight the value of the transmittance of the pump beam for the positive extrema of sensitivity.

The aim was to maximize the transmittance $T_{0, 3}^{L}$ of the bilayer {layer 1, layer 2} of the pump beam while, at the same time, being as sensitive as possible to the dynamic variations in thickness for the probe beam. Thus, transmittance is plotted in Fig. 4(b) as a function of d₂/λ_2,p (black line) for d₁ equal to 12 nm. Transmittance is maximum for every λ_2,p/2, and we can see that at maximum, less than half of the pump beam intensity is transmitted from the substrate to layer 3. In Fig. 4(b), indicated by the dotted line, we also present the sensitivity of the relative reflectivity change, $1 / R_{0, 3}^{L} \times \partial R_{0, 3}^{L} / \partial d_{2}$ ⁠, for d₁ = 12 nm as a function of d₂/λ_2,s. It can be seen that the extrema of the sensitivity for the probe beam do not match the maxima of the transmittance for the pump beam. Thus, in this case, it was not possible to maximize the amplitude of the generated acoustic waves and the sensitivity to the relative reflectivity change at the same time. Here, a larger gain in amplitude was expected by maximizing sensitivity to acoustic disturbance. Consequently, to maximize sensitivity and obtain the highest possible transmittance, the choice of the thickness of layer 2 was constrained to thicknesses $d_{2} \approx \frac{3}{4} λ_{2, s} + m λ_{2, s}$ with $m \in N$ ⁠. The first two values, d₂/λ_2,s = 0.75 and d₂/λ_2,s = 1.75, correspond to the two red dots with the highest possible values of $T_{0, 3}^{L}$ ⁠, as shown in Fig. 4(b).

In conclusion, to ensure the largest opto-acoustic transduction and an extremum of sensitivity to the acoustic signal at the same time, the thickness of the first layer was chosen to optimize the transmission coefficient of layer 1 to the pump and to enhance the sensitivity of the relative reflectivity change of the bilayer {layer 1, layer 2} to the probe. Multiple values of the thickness of the optical cavity are suitable to reach a compromise for high transmission of the pump beam in layer 3, as well as high sensitivity for the probe beam.

B. Simulation of dynamic reflectivity and comparison with a monolayer transducer

We completed the previous parametric analysis by calculating the dynamic reflectivity and the various acoustical echoes propagating in the structure. We also compared the three-layer transducer with a usual monolayer one. Both the distribution of the electromagnetic energy and dynamic reflectivity were calculated using the opto-thermo-elastic and elasto-optic models presented in Secs. II and III. The optical, thermal, and mechanical parameters considered for each layer are given in Table I.

We were interested in the acoustic echoes reflected at the interface between layer 3 and the sample on it as they carry information on sample impedance and adhesion. These echoes must not be overlapped with other echoes resulting from reflections at the multi-layer interfaces. The thicknesses of layers 2 and 3 must then be thoughtfully chosen to avoid echoes overlapping. Moreover, d₃ must be large enough to avoid any laser-induced damage, due to the temperature increase, to any sample placed on top of layer 3.¹⁶ The optimal set of layer thicknesses chosen was therefore d₁ = 12 nm, d₂ = 1021 nm, and d₃ = 300 nm.

In Fig. 5(a), we first compare the distribution of the electromagnetic density energy absorbed in the three-layer transducer Q_tri (red line) and in a 300 nm monolayer transducer Q_mono (blue line) sputtered onto a sapphire substrate for the same input fluency I₀ = 30 J m⁻². The density of energy of each structure was normalized by the maximum of the density of energy in the monolayer transducer.

FIG. 5.

(a) The density of electromagnetic energy in opto-acoustic transducers on a sapphire substrate as a function of depth z̆. The red line stands for the density of energy of a transducer made of a 12 m titanium layer, a 1021 nm LiF layer, and a 300 nm titanium layer (Qtri); the blue line stands for the density of energy of a monolayer of a 300 nm titanium transducer (Qmono). Both densities were divided by the maximum of the density of energy of the monolayer transducer. (b) The relative reflectivity change in a monolayer transducer composed of 300 nm titanium sputtered onto a sapphire substrate.

View large Download slide

(a) The density of electromagnetic energy in opto-acoustic transducers on a sapphire substrate as a function of depth $\overset{̆}{z}$ ⁠. The red line stands for the density of energy of a transducer made of a 12 m titanium layer, a 1021 nm LiF layer, and a 300 nm titanium layer (Q_tri); the blue line stands for the density of energy of a monolayer of a 300 nm titanium transducer (Q_mono). Both densities were divided by the maximum of the density of energy of the monolayer transducer. (b) The relative reflectivity change in a monolayer transducer composed of 300 nm titanium sputtered onto a sapphire substrate.

In the monolayer structure, the density of energy decreases exponentially with $\overset{̆}{z}$ and the rate of decrease depends on the optical absorption coefficient β_Ti,p = 1/ξ_Ti,p, as expressed by Eq. (3) for an opaque layer (b₃ = 0). The first layer of the three-layer transducer is spatially limited between $\overset{̆}{z} = 0$ and $\overset{̆}{z} = 12$ nm. The density of energy, plotted as a red line, increases exponentially with depth. This is due to the optical cavity effect in layer 2, which modifies the reflection coefficient of this layer, $r_{1, 3}^{L}$ ⁠, for the pump beam. The density of energy in layer 3, with a thickness of 300 nm, is visible from $\overset{̆}{z} = 1.034 μ$ m to greater distances; it decreases exponentially for the same reasons as for the monolayer transducer. We can see that the maximum density of energy in layer 3 is 2.3 times less than that in the monolayer transducer. Moreover, the maximum density of energy in layer 1 is 5 times greater than that in layer 3. This difference is due to the optical cavity effect of layer 2 for the pump beam (cf. Sec. II B).

In Fig. 5(b), we present the reflectivity change, ΔR_0,1/R_0,1, as a function of time for the 300 nm titanium monolayer transducer ΔR_0,1/R_0,1; thus, the density of energy is represented by a blue line in Fig. 5(a). The probe beam penetrates the titanium layer over a distance of less than half the optical wavelength λ_2,s. The relative reflectivity change in the layer reveals the acoustic waves that arrive below the surface. The arrivals of echoes reflected at the free surface of the transducer are clearly visible at 0.08 and 0.16 ns. The first echo corresponds to the acoustic wave propagating back and forth along the thickness of the layer and has an amplitude of ∼10⁻⁴. The second echo is reflected twice at the free titanium surface and once at the titanium–sapphire interface. The acoustic reflection coefficient for the acoustic strain at the titanium–vacuum interfaces is −1. Thus, the strain has a phase shift of π compared to the first echo. The decrease in amplitude of the echoes with time is due to acoustic attenuation and partial transmission of the acoustic waves into the substrate. The oscillations of smaller amplitude visible for the whole duration of the calculated signal are Brillouin oscillations and result from the interaction of the probe beam with the acoustic wave throughout its propagation in sapphire.

The previous analysis of the monolayer transducer is useful for understanding the intricate and complex variations in the relative reflectivity changes of the three-layer structure plotted in Fig. 6(a), where the time reflectivity response, $Δ R_{0, 3}^{L} / R_{0, 3}^{L}$ ⁠, was calculated for three different thicknesses of layer 2—d₂ = 1000 nm (gray dotted line), d₂ = 1019 nm (yellow), and d₂ = 1021 nm (dark blue line)—and with the same input power I₀, as that shown in Figs. 5(a) and 5(b).

FIG. 6.

View large Download slide

(a) Relative reflectivity change in the three-layer transducer deposited on a sapphire substrate. The transducer is composed of a LiF layer in between two titanium layers with thicknesses d₁ = 12 nm and d₃ = 300 nm. The calculations are for three values: d₂ = 1000 nm (gray dotted line), d₂ = 1019 nm (yellow), and d₂ = 1021 nm (turquoise blue line); the upright inset presents the path of the propagating acoustic waves for the echoes that reach the layer 2/layer 3 interface. The different paths are highlighted with red, blue, and gray arrows. (b) Normalized sensitivity of the relative reflectivity of the three-layered transducer calculated for d₁ = 12 nm vs the thickness of layer 2.

In order to determine the mechanical properties of a hypothetical sample in contact with the last layer of the transducer, we were interested in increasing the amplitude of the echoes that reflect at the last interface of the transducer. The upright inset in Fig. 6(a) represents the path of the propagating acoustic waves for these echoes that reach the layer 2/layer 3 interface at 0.08 and 0.24 ns. The first two different paths are highlighted with red and blue arrows, respectively. Echo 1 is generated in layer 3 and propagates back and forth in layer 3. Echo 2 is generated in layer 1, passes through layer 2, and bounces back and forth into layer 3. The last echo, indicated by the gray arrow, echo 1 → 2, is the acoustic wave generated in layer 1, which propagates from layer 1 to layer 3 through layer 2.

In terms of spectral components, we see that the reflectivity change presents a low-frequency contribution due to the thickness variations in the transparent LiF layer as a result of the acoustic wave. This results in steps on the low-frequency oscillations. Between 0 and 0.4 ns, high amplitude oscillations at frequency 35.2 GHz are also visible. They are due to Brillouin scattering in the thick transparent LiF layer. The oscillations of smaller amplitude at 74.7 GHz noticeable in the whole signal correspond to the detection of the acoustic wavefront propagating in the sapphire substrate.

The corresponding sensitivity of the opto-acoustic detection, $1 / R_{0, 3}^{L} \times \partial R_{0, 3}^{L} / \partial d_{2}$ ⁠, is presented in Fig. 6(b) as a function of the thickness of layer 2. Each thickness value, 1000 nm, 1019, or 1021 nm, is indicated by dots with the same color code as in Fig. 6(a). We can see that the yellow (d₂ = 1019 nm) and turquoise blue (d₂ = 1021 nm) curves shown in Fig. 6(a) correspond to the extrema of the sensitivity of the relative reflectivity change and that the gray dotted line (d₂ = 1000 nm) is associated with a lower sensitivity. The increase in amplitude between the gray dotted line and the yellow lines corresponds to the ten-fold gain in sensitivity from 0.08 for d₂ = 1000 nm to 0.8 for d₂ = 1019 nm [Fig. 6(b)]. This explains why the absolute value of the magnitude of both signals shown in Fig. 6(a) is greater than the reflectivity change for 1000 nm thickness. In Fig. 6(a) we also note that, due to the change in sign of the sensitivity shown in Fig. 6(b), the time oscillations associated with the low-frequency components for thicknesses of 1019 and 1021 nm are 180° phase shifted.

We observe that the absolute values of the amplitude of the echoes shown with gray and blue arrows are not exactly the same for the 1019 and 1021 nm graph lines. This is related to the fact that the change in sign of the sensitivity as d₂ increases from 1019 to 1012 nm is mainly due to the optical resonance of the optical cavity (layer 2). Meanwhile, the reflectivity change in layer 1 is smaller in amplitude, and its sign remains the same. However, we have seen that the dynamic reflectivity of the opto-acoustic detection in the three-layer transducer is the superposition of the contributions of the reflectivity of each layer. From d₂ = 1019 nm to d₂ = 1021 nm, the highest contribution (layer 2) is 180° phase shifted while the contribution of layer 1 is not. Consequently, the curves are not exactly opposite each other.

With Figs. 5(b) and 6(a), it is possible to compare the amplitude of the echo that merely goes back and forth in the 300 nm thick opaque titanium layer for both the monolayer and the three-layer transducers. The arrival time of both echoes is around 0.1 ns, but for the three-layer transducer, the echo is twice as large. Moreover, this increase in amplitude is produced at a lower density of energy. Indeed, Fig. 5(a) shows that the density of energy absorbed in layer 3 for the three-layer transducer is 2.3 times lower than the energy absorbed in the 300 nm thick monolayer transducer. Consequently, the sensitivity of the detection of the first echo for the three-layer transducer is nearly five times higher.

Echo 2, represented by the blue arrow, does not exist in the monolayer transducer because it is generated in the first thin titanium layer of the multi-layered transducer. As previously shown in Fig. 5(a), the density of energy is five times higher in this layer than the density of energy of the monolayer transducer. The generated acoustic wave is transmitted through the interface between the LiF layer and the 300 nm titanium layer before it is reflected and partially transmitted at the free surface. With an acoustic transmission coefficient $t_{2, 3}^{a c} =$ 0.7, the amplitude of echo 2 is then 3.5 times higher than that of echo 1, and the associated increase in amplitude is around seven times that of the amplitude of the first echo visible on the relative reflectivity change in the monolayer transducer.

We have shown that the intricate time variations in the dynamic reflectivity at the interface between layers 2 and 3 of the three-layer transducer are the result of the optical interaction of the probe beam with the acoustic waves generated in both titanium layers. These waves are reflected and transmitted multiple times at various interfaces of the transducer. The total dynamic reflectivity is then the superposition of the oscillations due to the Brillouin scattering in the transparent layers, the time steps caused by the modification of thickness due to the propagating acoustic waves, and the various echoes reflected and transmitted at the interfaces. The confrontation of the dynamic reflectivity with the sensitivity of the opto-acoustic detection has shown how the thickness of the optical cavity strongly influences the shape and amplitude of the dynamic reflectivity. Around the optical resonance, sensitivity is at its maximum, and the amplitude of the echoes that have encountered the last interface of the transducer is also maximized. We have shown that for equal absorbed energy density values, the three-layer transducer allows for five-fold and seven-fold increases in the amplitude of the first two echoes, respectively, compared to the amplitude of the echo generated in a monolayer transducer.

V. CONCLUSION

In this work, we designed a layered transducer composed of a transparent layer sandwiched between two metallic films to achieve efficient optical generation and optical detection of hypersounds, and we discussed the sensitivity improvement with respect to the monolayer transducer. For this, we calculated the transient reflectivity change in the structure induced by absorption of short light pulses. We pointed out that the combined change in the thicknesses of the semi-transparent and the transparent layers can be used to enhance the optical cavity effect in the transparent layer in order to maximize the relative reflectivity change and the sensitivity of the detection. At the same time, to ensure the greatest opto-acoustic transduction and an extremum of sensitivity to the acoustic signal, the thickness of the first layer was chosen to optimize the reflection and transmission coefficient of layer 1 and to enhance the sensitivity of the relative reflectivity change in the bilayer {layer 1, layer 2}. Multiple values for the thickness of the optical cavity, $d_{2} \approx \frac{3}{4} λ_{2, s} + m λ_{2, s}$ ⁠, were possible for the best compromise between the high transmission of the pump beam in layer 3 and the high sensitivity for the probe beam. In this work, an additional constraint, relating to the time of flight of the echoes generated in the structure, further restricted the choice of thickness for layer 2. In terms of echo amplitude, the deep understanding of the coupled optical and acoustical phenomena that influence dynamic reflectivity has allowed for the design of a three-layer transducer five to seven times more efficient than a comparable monolayer transducer at equal value of absorbed energy density. These results open large perspectives for the development of opto-acoustic transducers showing high conversion efficiency. Among the interest for an increase in the signal to noise ratio is the reduction of the acquisition time, as this will be greatly beneficial for the applications of the picosecond ultrasonics technique to industrial purposes and to the imaging of living matter.³

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Louise Le Ridant: Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (supporting); Software (lead); Validation (equal); Writing – original draft (lead); Writing – review & editing (equal). Marie-Fraise Ponge: Formal analysis (equal); Investigation (equal); Methodology (equal); Software (supporting); Supervision (supporting); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal). Bertrand Audoin: Conceptualization (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (lead); Project administration (equal); Supervision (lead); Validation (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

Dynamic reflectivity of an opto-acoustic three-layer transducer for picosecond ultrasonic measurements

I. INTRODUCTION

II. OPTO-ACOUSTIC GENERATION IN A MULTI-LAYERED STRUCTURE

A. Electromagnetic waves in the layers

B. Temperature increase in the layers

C. Acoustic waves in the multi-layered structure

III. ELASTO-OPTIC DETECTION IN A THREE-LAYER TRANSDUCER

A. Elasto-optic interaction with the acoustic strain inside each layer of the three-layer transducer

1. Reflectivity change of a transparent substrate and a transparent layer

2. Reflectivity change of the opaque layer

B. Reflectivity change due to interface motion

C. Dynamic reflectivity of the three-layer transducer

IV. SIMULATIONS AND RESULTS

A. Optical static reflectivity and transmission of a three-layer transducer

B. Simulation of dynamic reflectivity and comparison with a monolayer transducer

V. CONCLUSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Dynamic reflectivity of an opto-acoustic three-layer transducer for picosecond ultrasonic measurements

I. INTRODUCTION

II. OPTO-ACOUSTIC GENERATION IN A MULTI-LAYERED STRUCTURE

A. Electromagnetic waves in the layers

B. Temperature increase in the layers

C. Acoustic waves in the multi-layered structure

III. ELASTO-OPTIC DETECTION IN A THREE-LAYER TRANSDUCER

A. Elasto-optic interaction with the acoustic strain inside each layer of the three-layer transducer

1. Reflectivity change of a transparent substrate and a transparent layer

2. Reflectivity change of the opaque layer

B. Reflectivity change due to interface motion

C. Dynamic reflectivity of the three-layer transducer

IV. SIMULATIONS AND RESULTS

A. Optical static reflectivity and transmission of a three-layer transducer

B. Simulation of dynamic reflectivity and comparison with a monolayer transducer

V. CONCLUSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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