One of the main methods for obtaining information about the generation of sound pulses in metals is to measure the reflection coefficient of a probe wave. Various theoretical models are used to interpret the results of measuring the contribution to reflection coefficient $\Delta R(t)$ due to sound-generated displacements of lattice atoms. The purpose of this paper is to establish the degree of accuracy of models used in the case of sound generation in thin films exposed to a femtosecond pulse. It is shown below that the assumption of uniform heating used for thin films is justified if the film thickness is less than the film heating depth and for thicker films at times greater than the film heating time over the entire thickness. For optically thick films, a relatively simple expression for the field can be used. If the film thickness is less than the skin layer depth of the pump field, then it is necessary to consider the field reflection from a substrate. In this case, depending on the optical properties of the metal and the substrate, taking into account reflection can lead to either an increase or a decrease in $\Delta R(t)$. It has been established that if the skin layer at the frequency of probe radiation is less than the film heating depth, then taking into account temperature gradients in the equation for the displacement of lattice atoms leads to small changes in $\Delta R(t)$. This makes it possible to significantly simplify calculations of the displacement of lattice atoms.

## I. INTRODUCTION

Exposure of metals to femtosecond laser pulses is one of the main ways to produce picosecond pulses of sound. Interest in studying the generation of such pulses and their properties is associated with the possibility of their extensive practical application. Picosecond laser acoustics methods are used in studying the physical properties of materials,^{1–4} defects and cracks detection,^{5–8} samples structure examinations,^{9,10} and nanoscale objects diagnostics.^{11–13} Several works are devoted to the theoretical and experimental study of sound generation in thick metal films.^{14–19} Special attention is paid to sound generation in thin films, due to the possibility of generating sound in the terahertz and sub-terahertz frequency range.^{20–23}

In metals, the main cause of sound generation is a fast change in the temperatures of the lattice and electrons in space, leading to lattice deformation.^{14,15} In turn, the regularities of temperature changes depend on the laser field structure in the metal, its thermal conductivity, the rate of energy exchange between electrons and the lattice, and the heat capacity of the electrons and the lattice. For films used in experiments, the effect of their thickness on heat capacity and energy transfer is usually insignificant. On the contrary, the structure of the electromagnetic field that heats electrons changes significantly with the film thickness, and also depending on a substrate material. The degree of temperature inhomogeneity, which depends on the metal’s thermal conductivity, also changes significantly with changes in film thickness. Therefore, sound generation in the film has features associated with its finite thickness. Examination of these features is the purpose of this communication. One of the common methods for studying sound in metals is to measure $\Delta R(t)$—the reflection coefficient change of a sample under conditions of its heating by a femtosecond pulse of electromagnetic radiation. In this regard, the main attention is paid to describing those changes in the reflection coefficient that are caused by the finite dimensions of the film. When describing the reflection of probe radiation, the effect of film thickness on its penetration into the metal is also taken into account.

The consideration is based on equations for the displacement of lattice atoms and equations for the electron and lattice temperatures. The heating of electrons in the film is described taking into account the field distribution obtained by solving Maxwell’s equations in the film and substrate. Using these equations, the Laplace images of electron and lattice temperature increments are found, as well as $ u z(z,\omega )$—the Laplace image of the displacement of lattice atoms. The resulting expression for $ u z(z,\omega )$ was used to analyze changes in the reflection coefficient $\Delta R(t)$ arising due to the displacement of lattice atoms. It is shown how temperature inhomogeneity across the film thickness affects sound generation and associated changes in $\Delta R(t)$. An analysis of the influence of electromagnetic field structure on $\Delta R(t)$ is presented. It is shown how the reflection of the electromagnetic field at the metal–substrate interface changes $\Delta R(t)$, and how these changes depend on the optical properties of the film and substrate. Conditions have been found under which it is possible to approximately ignore the influence of temperature gradients inside the film on the displacement of atoms and the associated change in the metal reflectance.

## II. BASIC EQUATIONS

^{24,25}

^{14,15}We restrict ourselves to consider times shorter than the time it takes for sound to travel through the thickness of the substrate. That is, the substrate is quite thick. When describing sound generation, we use the equation for the component of lattice displacement vector $ u z(z,t)$ along the direction of temperature inhomogeneity

^{14,15}

## III. LAPLACE IMAGE OF THE DISPLACEMENT OF LATTICE ATOMS

## IV. INFLUENCE OF INHOMOGENEOUS HEATING

As can be seen from Eq. (5) and the expression for $\sigma (z,\omega )$, in the frequency range $\omega <\gamma G/ \gamma e C e$ the main cause for sound generation is the inhomogeneity of the lattice temperature. For typical metals, such frequencies are less than one terahertz. Since the lattice is heated due to the transfer of energy from electrons, inhomogeneity of $\Delta T$ arises if the electron temperature is inhomogeneous. In the absence of heat flux at the film boundaries, $\Delta T e$ changes abruptly at the film boundaries. Therefore, one of the reason for sound generation is the jump in $\Delta T e$, and thereby $\Delta T$, at $z=0,L$. Generation is possible throughout the entire film thickness until the temperature of inhomogeneously heated electrons is equalized across the film thickness. In accordance with Eq. (2), the leveling time of $\Delta T e$ is of the order of $ C e L 2/\lambda =3 \nu s L 2/ v F 2$. If this time is shorter than $ C e/G$—the cooling time of electrons due to transferring energy to the lattice, then before $\Delta T$ changes, the temperature will become uniform over the film thickness. That is, for $L< \lambda / G$, the influence of $\Delta T$ inhomogeneity on sound generation will become insignificant. On the contrary, for $L> \lambda / G$, such a cause of generation must be taken into account until the time moment $\u223cC L 2/\lambda $. Note that such a generation pattern is possible only in films whose thickness $L$ is greater than $1/| \kappa L|$—the penetration depth of laser radiation that heats the electrons in the metal. If $| \kappa L|L<1$, then the film is heated uniformly. To illustrate the above qualitative considerations, this section shows how changing the film thickness changes the reflection coefficient of the probe wave of a silver film exposed to a femtosecond pulse, the absorption of which is accompanied by sound generation.

^{26}is the concentration of lattice atoms, $m\u2248 m e$,

^{27}

^{,}$ m e$ is the electron mass, $ \omega p\u22481.37\xd7 10 16$ s $ \u2212 1$, $ \nu s\u22484.5\xd7 10 13$ s $ \u2212 1 $,

^{28}

^{,}$C=3 k BN\u22482.4\xd7 10 7$ erg/cm $ 3$ K, $ k B$ is the Boltzmann constant, $ C e\u22482.0\xd7 10 5$ erg/cm $ 3$ K,

^{29}

^{,}$G\u22483.5\xd7 10 17$ erg/cm $ 3$ Ks,

^{29}

^{,}$ v F=1.4\xd7 10 8$ cm/s,

^{26}

^{,}$\gamma =2.3$ and $ \gamma e=1.2$,

^{30}

^{,}$\rho =10.5$ g/cm $ 3$ and $ v l=3.7\xd7 10 5$ cm/s.

^{31}As the substrate we use SiO $ 2$, in which $ \epsilon d=2.1$, $ \rho d=2.7$ g/cm $ 3$ and $ v d=5.7\xd7 10 5$ cm/s.

^{31}In this case $\mu =0.39$. We also assume that $ E L(t)= E Lexp[\u2212 t 2/2 t p 2]$ and $ t p=50$ fs, and the peak energy flux density of the pump pulse $I=c E L 2/8\pi =2\xd7 10 8$ W/cm $ 2$. Let us make a comparison at $\u210f \omega 0=1.5$ eV. At this frequency, $\nu =1.9\xd7 10 14$ s $ \u2212 1$ and $\epsilon ( \omega 0)=\u221229+0.86i$.

^{28}For the above parameters, the film heating depth is $ \lambda / G\u224890$ nm, and the attenuation depth of the electromagnetic field is $\u224824$ nm. We also assume that the frequency of probe radiation does not differ from the frequency of pump wave and $\u210f \omega p r=1.5$ eV. In calculations, we use the relation

In Fig. 2, solid curves show the results of calculations at $\u210f \omega 0=1.5$ eV using the expressions (11) and (12). As can be seen from the figure, the $\Delta R(t)/R$ plots represent a set of broadened peaks that appear around a certain shifted position. This form of the plots can be explained by the fact that there are two contributions to $\eta (z,\omega )$. The first of them corresponds to sound pulses arising at the film boundaries due to a jump in lattice temperature. This contribution is associated with peaks on the $\Delta R(t)/R$ profile, which, in the case of $L=50$ nm, occur at $t=nL/ v l\u224813.5n$ ps, and in the case of $L=200$ nm at $t\u224854n$ ps, where $n=0,1,2,\u2026$. There are two sets of peaks. The first set of peaks, at times around $t=2nL/ v l$, has a larger amplitude and corresponds to sound arising at the metal–vacuum interface. The second set, around $t=(2n+1)L/ v l$, has a smaller amplitude and is associated with sound arising at the metal–dielectric interface. With each reflection of sound from a dielectric, its amplitude decreases $(1+\mu )/(1\u2212\mu )\u22482.3$ times, which is due to the partial transmission of sound into the dielectric. Note that for sufficiently thick films, in which heating of the film at the metal–dielectric interface is negligible, the second set of peaks will be absent (see Ref. 19 for more details). The second contribution to $\eta (z,\omega )$ is associated with the presence of a temperature gradient across the entire film thickness. As the temperature equalizes due to heat transfer by electrons, this contribution to sound generation decreases. Over time, the temperature becomes homogeneous throughout the film thickness, and a constant displacement of $\eta (z,\omega )$ is responsible for the thermal expansion of the film due to heating. It is this displacement that leads to the existence of a shifted position on the $\Delta R(t)/R$ plots (see Fig. 2).

When studying sound generation in thin films, it is sometimes approximately assumed (see, for example, Refs. 21, 22) that the electron temperature is uniform throughout the film thickness. It is of interest to determine the degree of accuracy of such an approximation. For this purpose, in Fig. 2, the dashed curves show the results of calculations of $\Delta R(t)/R$ using analogs of expressions (11) and (12) obtained for the case of uniform heating of the film. Such expressions can be obtained by passing to the limit $\lambda \u2192 \u221e$ in (11) and (12). At $L=50$ nm, the electron temperature equalization time $ C e L 2/\lambda \u22480.17$ ps is less than their cooling time $ C e/G\u22480.54$ ps and earlier than $\Delta T$ changes, the temperature becomes uniform throughout the film thickness. In this case, as can be seen from Fig. 2(a), the solid and dashed curves are almost indistinguishable. On the contrary, at $L=200$ nm, the temperature equalization time is $\u22480.68$ ps and is longer than $0.54$ ps, and the influence of inhomogeneous heating on sound generation is no longer small. In this case, the solid and dashed curves in Fig. 2(b) differ noticeably until time $C L 2/\lambda \u2248350$ ps. From the presented comparison, it is clear that a good criterion for the applicability of using the uniform temperature approximation to describe sound generation in thin films is the inequality $L< \lambda / G$. In thicker films, such an approximation is justified after the heating time of the film over the entire thickness has elapsed, which is $\u223cC L 2/\lambda $.

## V. INFLUENCE OF THE ELECTROMAGNETIC FIELD STRUCTURE

Let us discuss to what extent the dependence of the field structure on the film thickness affects sound generation. Above, we considered the change in the reflection coefficient of the silver film upon excitation of sound by a femtosecond radiation pulse with a quantum energy of $\u210f \omega 0=1.5$ eV. As already noted, such radiation penetrates into the silver to a depth of $\u223c1/ \kappa L \u2032\u224824$ nm. If the film thickness is greater than the penetration depth, that is, the inequality $ \kappa L \u2032L\u226b1$ is satisfied, then expressions for the field in the half-space can be used to describe the heating of the film and the subsequent excitation of sound. In this case, the displacement of lattice atoms is described by simpler formulas 14 and 16 from Ref. 19. The degree of accuracy of this approximation is illustrated in Fig. 3, which shows plots of $\Delta R(t)/R$ obtained by using these formulas and expressions (11) and (12), taking into account the influence of finite film thickness on the field structure. The graphs are plotted for a film with a thickness of $L=50$ nm. As can be seen from Fig. 3(a), even when the field penetration depth is slightly exceeded, the approximate and exact curves are in good agreement. Good accuracy is achieved due to the exponentially rapid decrease in the field deep into the thick film.

The situation changes if $ \kappa L \u2032L\u226a1$. For example, at $\u210f \omega 0=1.5$ eV and $L=10$ nm, taking into account the reflection of the electromagnetic field from the metal–dielectric surface leads to the increasing both amplitude of sound waves and displacement of atoms due to the film thermal expansion. More efficient excitation of sound occurs due to a relative increase in the field strength inside the film. The reason for this enhancement can be understood if we note that at $\u210f \omega 0=1.5$ eV the reflection coefficient of radiation from the Ag–SiO $ 2$ interface is close to unity: $ r A g \u2212 S i O 2=( \epsilon ( \omega 0 )\u2212 \epsilon d)/( \epsilon ( \omega 0 )+ \epsilon d)\u22481$. Since $ \kappa L \u2033L\u226a1$, the interference of the penetrating and reflected fields occurs in phase, which leads to an increase in the field inside the film. This manifests itself in a relative increase in reflectance [see Fig. 3(b)]. In Fig. 3(b), along with the plots of $\Delta R(t)/R$ obtained by using formulas (11) and (12), the dashed curve shows the data obtained without taking into account the effect of the finite film thickness on the field strength.

The influence of field reflection from the metal–dielectric interface may be different. Let us demonstrate this in the case of exposure of a film with a thickness of $L=50$ nm to radiation with quantum energy of $\u210f \omega 0=3.9$ eV and a frequency of $ \omega 0=5.9\xd7 10 15$ s $ \u2212 1$ close to the boundary of interband transitions. In this case, $\epsilon ( \omega 0)=0.74+0.70i$,^{28} and the field penetration depth increases significantly up to $\u2248180$ nm. Note that in this case, the effective frequency of electron collisions also increases up to $\nu =1.1\xd7 10 15$ s $ \u2212 1$.^{28} In this case, the permittivity of the substrate did not change. Due to the large change in the permittivity of silver, the reflection coefficient from the Ag–SiO $ 2$ interface also changes. The real part $ r A g \u2212 S i O 2$ changes sign, and $ \kappa L \u2033L$ is still less than one. As a result, the interference of the field penetrating into the film and reflected from the Ag–SiO $ 2$ boundary occurs under conditions when their phase is shifted by more than $\pi /2$. The result of such interference is a weakening of the total field compared to the case when reflection from a distant film boundary is not taken into account. As a result, the difference between the solid and dashed curves in Fig. 4 turns out to be exactly the opposite of what is shown in Fig. 3(b). At the same time, despite the negative influence of reflection from the metal–dielectric interface, the absolute values of the change in the reflection coefficient in Fig. 4 are significantly larger than in Fig. 3(a). This is because at the frequency $ \omega 0=5.9\xd7 10 15$ s $ \u2212 1$ the frequency of electron collisions has greatly increased. The field enhancement from increasing field depth did not appear because of the relative attenuation due to reflection from the substrate.

## VI. THE INFLUENCE OF TEMPERATURE GRADIENTS

Let us use these expressions to calculate $\Delta R(t)/R$. In Fig. 5, the dashed curve corresponds to the calculation using formulas (16) and (17). The solid curve is the same as in Fig. 5(b). The calculation was performed for probe radiation with $\u210f \omega 0=1.5$ eV and the film with a thickness of $L=200$ nm. Both curves in Fig. 5 are pretty close to each other. The reason for the closeness of the $\Delta R(t)/R$ curves can be explained as follows. The change in the reflection coefficient at the frequency of probe radiation occurs due to the displacement of atoms inside the skin layer with a thickness of $1/ \kappa L \u2032\u224824$ nm. At the same time, already from the moment $\u22483 \nu s/ ( \kappa L \u2032 v F ) 2=40$ fs, temperatures change on scales greater than the skin layer depth. That is, temperatures inside the skin layer depend negligibly on $z$ at the times under consideration, and simpler expressions (16) and (17) can be used to calculate $\Delta R(t)/R$). If the skin layer depth at the frequency of probe radiation is greater than or comparable to the scale of temperature changes, then temperature gradients can affect $\Delta R(t)/R$.

## VII. CONCLUSION

Above, laser generation of sound in a metal film on a dielectric substrate was studied, taking into account both the finite rate of heat transfer across the film thickness and the exact structure of the electromagnetic field inside the metal. Expressions for changes in the temperature of the lattice and electrons are obtained, and by using them, the Laplace image of the displacement of lattice atoms arising due to temperature changes is found. To understand the degree of influence of the thermal conductivity of the metal and the electromagnetic field reflection from the rear surface of the film, a comparison with the results for a uniformly heated film and an optically thick film is given. For this purpose, we considered the metal reflectivity change caused only by lattice deformation, without taking into account such changes due to heating of electrons and their photoexcitation. Corresponding plots for $\Delta R(t)/R$ are compared in various cases, obtained using exact and approximate expressions. The difference between the exact result and the approximation of a uniform temperature appears at a film thickness greater than the metal heating depth. With such thicknesses, with an approximate description, the amplitude of sound pulses arising at the metal–vacuum boundary turns out to be smaller, and at the rear surface, correspondingly, greater than with an accurate one. Also, the homogeneous temperature approximation does not take into account the relaxation of the contribution from the temperature gradient to $\Delta R(t)/R$. In turn, for thick films, the thickness of which is greater than the skin layer depth, the optically thick film approximation works well. However, if the film thickness turns out to be less than the penetration depth of the heating field, then it is necessary to take into account the field reflection from the metal–dielectric boundary. In this case, depending on the optical properties of the metal and dielectric at the frequency of the heating pulse, taking into account reflection can lead to both an increase in the displacement of atoms and a decrease in them. Finally, the effect of taking temperature gradients into account in the equation for the displacement of lattice atoms on the reflectance change is considered. It has been established that if the skin layer depth at the frequency of probe radiation is less than the film heating depth, their influence is small.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

All authors contributed equally to this work.

**E. A. Danilov:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). **S. A. Uryupin:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

### APPENDIX: DISPLACEMENT OF LATTICE ATOMS IN A DIELECTRIC

## REFERENCES

*Introduction to Solid State Physics*

*CRC Handbook of Chemistry and Physics*