**2. Laser-driven acoustic waves in thin metal foils**

The interaction of pulse laser beam with metal surface is very complex phenomenon but our specific interest is in formation of the acoustic waves in irradiated material. To generate an acoustic wave a time dependent stress needs to be applied to the solid. A laser pulse is an excellent tool to generate this kind of the stress. There are two principal mechanisms of laser-induced stress formation in the solids: a) thermal stress, resulted from the non-uniform heating of the irradiated surface by the laser beam; b) mechanical stress due to mechanical impulse transferred from the leaving plasma plume formed on the surface during laser ablation. The parameters of the acoustic waves generated for these two processes are slightly different and will be discussed in details later. Here we will use acoustic wave theory but one should note that its' use is applicable only when the magnitude of the applied stress is small in comparison with the Young's modulus of the material. Large applied stresses can cause the development of the shock waves a phenomenon with different characteristics than acoustic waves (Menikoff 2007). Shock waves have been also hypothesized to be the driver of the LIAD phenomenon, and, as such cannot be entirely excluded from consideration, especially in some extreme cases. Nevertheless, during the last decade, laser-driven acoustic waves emerged in the literature as the "prime suspect" in the LIAD case.

### **2.1 Acoustic waves in metal foils**

The general governing equation for the generation of elastic waves in solids can be derived by combining the equation of motion and the Hooke's law. In general case it is a differential tensor equation, which interconnects the stress tensor, applied to the body, the displacement of the body's elemental volumes (strain), and their elastic properties. In order to analyze the data in detail, the appropriate stress tensor needs to be determined, and the corresponding set of partial differential equations for strain and stress must be resolved (Pollard 1977). One can simplify this analysis by taking into account specifics of the experiments. For thin foils with *h/R0*<<1 (where *R0* is the radius of the target foil, and *h* is its thickness), a round thin plate approximation can be applied to describe and analyze this problem (Smith 2000). The rise of the strain due to laser heating and the consequent development of the plasma plume can be considered as an external driving force. Generally speaking, the thin plate equation is a differential equation of the forth order,

$$\left(\frac{\partial^2}{\partial r^2} + \frac{1}{r} \cdot \frac{\partial}{\partial r} + \frac{1}{r^2} \cdot \frac{\partial^2}{\partial \theta^2}\right)^2 \tilde{\xi} + v\_L^{-2} \cdot \left(2 \cdot \alpha \frac{\partial \tilde{\xi}}{\partial t} - \frac{\partial^2 \tilde{\xi}}{\partial t^2}\right) = F(r, t) \tag{1}$$

where *ξ=ξ(r,t)* is the surface displacement in the *z* direction (perpendicular to the sample surface), *F(r,t)* is the external driving force caused by the laser irradiation, *vL* is a parameter depending on the material density, *ρ*, thickness, *h*, and flexural rigidity, *D*. Furthermore,

$$w\_{\perp} = \sqrt{D \,/\, \rho \cdot h} \quad \text{and} \quad D = \left(N \cdot h^{\dagger} \,/\, 12 \cdot (1 - \varepsilon^{2})\right) \tag{2}$$

where ε is the Poisson ratio, *N* is Young's modulus and α is the oscillation decay constant. A vibrating thin plate is one of the most frequently analyzed mathematical problems, and its detailed analysis can be found elsewhere (McLachlan 1951; Smith 2000). In the discussion given below, we will remain in the framework of the analysis given by (Smith 2000).

Molecular Desorption by Laser–Driven

having the form

right side of it.

Eq.(9) over *n*.

*t<*τ

τ

Acoustic Waves: Analytical Applications and Physical Mechanisms 347

Using Eq. (7), Eq. (1) can be split into two independent equations, with the equation for *ξ(t)*

 ξ α

2 <sup>2</sup> −⋅⋅ + ⋅= 2 ( ) *<sup>n</sup> d d F t*

2 2

 τ  ττ

, (8)

, (9)

, (10)

 ωξ

and the equation for *g(r)* being a non-uniform Bessel equation with the *δ(r)*- function in the

The exact solution of Eq. (8) can be expressed as the convolution of the Green function of the problem (8) and real time shape of *F(t)*. The governing equation for the Green function will be Eq.(8) with *δ*-function in the right side. It can be easily derived by applying the Laplace transform to the Eq.(8), subsequently solving the obtained linear equation in the *s*-space and returning back to the time space with using the inverse Laplace transform. As a result of

> 2 2 ( ) sin ( ) () −⋅− = ⋅ − ⋅− ⋅ <sup>−</sup>

*<sup>e</sup> <sup>t</sup> t Fd*

and the complete solution of Eq.(8) can then be obtained as the sum of the components of

Thus, the generation of acoustic waves in thin foils can be described by Eq.(9) which strongly depends on driving force *F(t)* after whose cease the vibration evolves into decaying harmonic oscillations (Eq.(4)) with frequencies defined by Eq.(6). It is apparent that maximal surface velocities can be achieved only at the initial stage of acoustic wave generation when

. The application of Eq.(9) to analysis of laser-driven acoustic vibrations is complicated by the lack of the exact knowledge of the time profile of the driving force *F(t)*. Depending on regime of the surface irradiation, this force may be of different origins and, accordingly, have strongly varying magnitudes and time profiles. The appropriate mechanisms will be

**2.2.1 The action of laser pulse on the metal surface: heating and plasma generation**  Metals subjected to pulsed laser irradiation absorb energy within a very thin surface layer (the skin-depth for most metals is less than 10-7 m) so that the temperature of the irradiated surface can rise extremely fast. For moderate laser intensities (below the plasma formation threshold) and a Gaussian-shaped laser beam, the maximum surface temperature can be

> max 1/2 (0) 2.15 ( ) <sup>=</sup> <sup>Κ</sup> *AI <sup>T</sup>*

where *Tmax(0)* is the maximum surface temperature at *z*=0 (the *z* direction is orthogonal to the target surface), *A* is the laser radiation absorption coefficient, *Imax* is the peak laser power,

After cessation of the laser pulse, the adsorbed energy continues to diffuse into the bulk metal and along the metal surface, resulting in a temperature increase underneath the

is the laser pulse duration, and *c, ρ,* and *Κ* are the specific heat, density and thermal

*c* τ

π ρ

1/2 max

ω α

2

these procedures, one can finally obtain the following equation

*t t*

ω α

α τ

*o n*

**2.2 Generation of the acoustic waves by laser pulses** 

conductivity of the corresponding metal, respectively.

estimated using a well-known expression (Prokhorov, Konov et al. 1990)

ξ

discussed in the following section.

( )

*n n*

ξ

*dt dt*

Assuming separate solutions for the radial and tangential terms, Eq. (1) can be converted into a system of three differential equations of the second order. Being primarily interested in the foil displacement in normal (*z* ) direction to the surface at the epicenter (*r*=0) and assuming that the external driving force *F(t)* is a pulse function lasting a specific time *τ,* we can separate the variables in the Eq. (2) for *t>>τ*. Under the assumption of harmonic motion for all modes, the surface displacement can be expressed as , <sup>1</sup> (,) () − ⋅ = ⋅ *m n i tt rt r e* ω α ξ ξ , where *ωm,n* is the vibration frequency. The general governing equation can then be written in the following form:

$$
\left(\frac{\partial^2}{\partial r^2} + \frac{1}{r} \cdot \frac{\partial}{\partial r} - \frac{n^2}{r^2} \pm k\_{n,w}^2\right) \tilde{\xi}\_1 = 0 \tag{3}
$$

where *n* is an integer number, *k v nm nm L* , , = ω / and *vL* is the wave velocity as in Eq.(2). Under the assumption that the oscillations are harmonic and decay exponentially, the following solution for the surface displacement can be obtained (McLachlan 1951):

$$\mathcal{L}\_n(r,t) = \mathcal{\xi}\_0 \cdot \left( I\_n(k\_{n,w}r) + \mathcal{X} \cdot I\_n(k\_{n,w}r) \right) \cdot \exp(i\alpha\_n t) \cdot \exp(-\alpha \cdot t) \tag{4}$$

where *Jn* and *In* are Bessel functions and *χ* is a constant. The term describing the angular dependence of the oscillations is omitted in Eq.(4). The fact that foils used in LIAD experiments are typically glued or welded on their perimeter corresponds in our analysis to the situation when edges of the round plate are fixed (i.e. non-vibrating), and is described by the following boundary conditions:

$$
\xi(R\_0, t) = 0, \frac{d\tilde{\xi}(R\_0, t)}{dt} = 0 \,\,\,\,\,\tag{5}
$$

that lead to an equation, whose solutions *jn,m* have tabulated values (Smith 2000). The corresponding vibration frequencies, *ωn,m*, can be then expressed as

$$\alpha\_{n,m} = \sqrt{j\_{n,m}^4 \cdot \frac{\upsilon\_{\perp}^2}{R\_0^4} - \alpha^2} = \frac{j\_{n,m}^2 \cdot \hbar}{R\_0^2} \cdot \sqrt{\frac{N}{12\,\rho \cdot (1 - \varepsilon^2)}}\tag{6}$$

Because values of α are small compared to the first term under the square root sign in Eq.(6), it is a reasonable assumption that the frequency is proportional to the square root of the ratio of Young's modulus, *N*, to the density of the foil material, *ρ*. As described by Eq. (6), in the steady-state regime (driving force *F(t)*=0 for *t>>τ*) the frequency spectra and decay times of the oscillation will remain the same while the laser intensity is varied, and only the amplitude should change.

For short times, (*t<*τ), the approach to solving Eq.(1) is to assume that the external force is a delta-function in space *F(r,t)=F(t)δ(r)* (point source)*,* allowing the solution of Eq.1 to be expressed in the form

$$
\tilde{\boldsymbol{\xi}}(\boldsymbol{r},t) = \boldsymbol{g}(\boldsymbol{r}) \cdot \tilde{\boldsymbol{\xi}}(t) \tag{7}
$$

where *g(r)* and ξ*(t)* represent the spatial and the time dependencies of the final solution, respectively. This approximation should help to develop a clearer understanding of physical problems related to the laser generation of acoustic waves in thin foils.

Assuming separate solutions for the radial and tangential terms, Eq. (1) can be converted into a system of three differential equations of the second order. Being primarily interested in the foil displacement in normal (*z* ) direction to the surface at the epicenter (*r*=0) and assuming that the external driving force *F(t)* is a pulse function lasting a specific time *τ,* we can separate the variables in the Eq. (2) for *t>>τ*. Under the assumption of harmonic motion

*ωm,n* is the vibration frequency. The general governing equation can then be written in the

2 2 , 1 <sup>1</sup> <sup>0</sup> ∂ ∂ +⋅ − ± = ∂ ∂ *n m*

Under the assumption that the oscillations are harmonic and decay exponentially, the

*n n* ( , ) ( ) ( ) exp( ) exp( ) = ⋅ + ⋅ ⋅ ⋅ −⋅ 0, , ( ) *<sup>n</sup> m n <sup>n</sup> m n*

where *Jn* and *In* are Bessel functions and *χ* is a constant. The term describing the angular dependence of the oscillations is omitted in Eq.(4). The fact that foils used in LIAD experiments are typically glued or welded on their perimeter corresponds in our analysis to the situation when edges of the round plate are fixed (i.e. non-vibrating), and is described by

> ( ,) ( , ) 0, 0 = = *d Rt R t dt* ξ

that lead to an equation, whose solutions *jn,m* have tabulated values (Smith 2000). The

, , 42 2

it is a reasonable assumption that the frequency is proportional to the square root of the ratio of Young's modulus, *N*, to the density of the foil material, *ρ*. As described by Eq. (6), in the steady-state regime (driving force *F(t)*=0 for *t>>τ*) the frequency spectra and decay times of the oscillation will remain the same while the laser intensity is varied, and only the

delta-function in space *F(r,t)=F(t)δ(r)* (point source)*,* allowing the solution of Eq.1 to be

respectively. This approximation should help to develop a clearer understanding of physical

 ξ

ξ

problems related to the laser generation of acoustic waves in thin foils.

<sup>⋅</sup> = ⋅−≈ ⋅ ⋅ − *L n m*

*v N j h <sup>j</sup> R R*

2 2 4 2 ,

 α 0

0 0 12 (1 )

are small compared to the first term under the square root sign in Eq.(6),

), the approach to solving Eq.(1) is to assume that the external force is a

*(t)* represent the spatial and the time dependencies of the final solution,

ρ

*<sup>n</sup> <sup>k</sup>*

2

ξ

<sup>1</sup> (,) () − ⋅ = ⋅ *m n i tt rt r e* ω α

, where

(3)

 ξ

 α

ξ

/ and *vL* is the wave velocity as in Eq.(2).

 ω

, (5)

ε

(,) () () *rt gr t* = ⋅ (7)

(6)

*rt J k r I k r i t t* (4)

for all modes, the surface displacement can be expressed as ,

2 2

*r rrr*

following solution for the surface displacement can be obtained (McLachlan 1951):

 χ

ω

0

ξ

corresponding vibration frequencies, *ωn,m*, can be then expressed as

*nm nm*

ω

following form:

where *n* is an integer number, *k v nm nm L* , , =

ξξ

the following boundary conditions:

α

τ

ξ

Because values of

amplitude should change. For short times, (*t<*

expressed in the form

where *g(r)* and

Using Eq. (7), Eq. (1) can be split into two independent equations, with the equation for *ξ(t)* having the form

$$\frac{d^2\xi}{dt^2} - 2\cdot\alpha \cdot \frac{d\xi}{dt} + \alpha\_n^2 \cdot \xi = F(t) \, \, \, \, \, \tag{8}$$

and the equation for *g(r)* being a non-uniform Bessel equation with the *δ(r)*- function in the right side of it.

The exact solution of Eq. (8) can be expressed as the convolution of the Green function of the problem (8) and real time shape of *F(t)*. The governing equation for the Green function will be Eq.(8) with *δ*-function in the right side. It can be easily derived by applying the Laplace transform to the Eq.(8), subsequently solving the obtained linear equation in the *s*-space and returning back to the time space with using the inverse Laplace transform. As a result of these procedures, one can finally obtain the following equation

$$\xi\_n(t) = \int\_{\circ} \frac{e^{-\alpha \cdot (t-\tau)}}{\sqrt{\alpha\_n^2 - \alpha^2}} \cdot \sin\left[\sqrt{\alpha\_n^2 - \alpha^2} \cdot (t-\tau)\right] \cdot F(\tau)d\tau \,, \tag{9}$$

and the complete solution of Eq.(8) can then be obtained as the sum of the components of Eq.(9) over *n*.

Thus, the generation of acoustic waves in thin foils can be described by Eq.(9) which strongly depends on driving force *F(t)* after whose cease the vibration evolves into decaying harmonic oscillations (Eq.(4)) with frequencies defined by Eq.(6). It is apparent that maximal surface velocities can be achieved only at the initial stage of acoustic wave generation when *t<*τ. The application of Eq.(9) to analysis of laser-driven acoustic vibrations is complicated by the lack of the exact knowledge of the time profile of the driving force *F(t)*. Depending on regime of the surface irradiation, this force may be of different origins and, accordingly, have strongly varying magnitudes and time profiles. The appropriate mechanisms will be discussed in the following section.

### **2.2 Generation of the acoustic waves by laser pulses**

### **2.2.1 The action of laser pulse on the metal surface: heating and plasma generation**

Metals subjected to pulsed laser irradiation absorb energy within a very thin surface layer (the skin-depth for most metals is less than 10-7 m) so that the temperature of the irradiated surface can rise extremely fast. For moderate laser intensities (below the plasma formation threshold) and a Gaussian-shaped laser beam, the maximum surface temperature can be estimated using a well-known expression (Prokhorov, Konov et al. 1990)

$$T\_{\text{max}}(0) = 2.15 \frac{A I\_{\text{max}} \pi^{1/2}}{\left(\pi c \rho \mathbf{K}\right)^{1/2}},\tag{10}$$

where *Tmax(0)* is the maximum surface temperature at *z*=0 (the *z* direction is orthogonal to the target surface), *A* is the laser radiation absorption coefficient, *Imax* is the peak laser power, τ is the laser pulse duration, and *c, ρ,* and *Κ* are the specific heat, density and thermal conductivity of the corresponding metal, respectively.

After cessation of the laser pulse, the adsorbed energy continues to diffuse into the bulk metal and along the metal surface, resulting in a temperature increase underneath the

Molecular Desorption by Laser–Driven

may be expressed as

thermal expansion coefficient.

of the original results.

Acoustic Waves: Analytical Applications and Physical Mechanisms 349

accordance with the general theory of thermal stresses in thin plates (Boley and Weiner 1960), a non-uniform heating of the surface is equivalent to a *negative loading pressure* and

> /2 2

*T r*

η

ε

by both the temperature profile (Eq. (11)) and by elastic properties of the material.

**2.2.3 Experimental observations of laser-generated acoustic waves in thin foils**  Experimental studies of acoustic waves in solids due to pulsed laser irradiation have started with the advent of such lasers (White 1963). A great collection of experimental results and theoretical analyses of acoustic wave generation in solids driven by laser pulses has been accumulated since (Hutchins 1985), and these studies continue at present (Xu, Feng et al. 2008). Regrettably, there is a very limited data set, which could be used to interpret of LIAD experiments. To prove (or disapprove) the "shake-off" hypothesis of molecular desorption, direct measurements of thin foil surface velocities in back-side irradiation geometry are required. The scarcity of such data motivated us to setup a series of our own experiments aiming at measurements of thin foils vibrations under typical LIAD conditions. Experimental approaches to this problem are well known and described in the literature (Scruby and Wadley 1978; Royer and Dieulesaint 2000). Nevertheless, we will briefly describe below our system, in order to create a better stage for presentation and discussion

One of the most popular and widely used methods to studies of acoustic waves is based on non-contact optical measurements. Figure 1a shows the experimental setup for measurements of surface displacement using interferometry-based approach. A He-Ne laser (1) (Melles-Griot, 543 nm, 0.5 mW) was used as the light source for a Michelson interferometer. It consisted of a beam splitter (5), an etalon and steering mirrors (3, 4), a focusing lens (6), a target (7), an imaging lens (9), an aperture (13), a focusing lens (14) and a photomultiplier (15). The target was back-irradiated by a pulsed laser (12) through a fused silica lens (8). Laser beam parameters were measured by intersecting the laser flux with two

**2.2.3.1 Experimental technique: optical and electrical methods** 

/2 (,) <sup>1</sup> <sup>−</sup> <sup>⋅</sup> = − ⋅∇ <sup>⋅</sup> <sup>−</sup> *h*

, (14)

*h <sup>N</sup> P T <sup>r</sup> <sup>z</sup> zdz*

where the temperature distribution over *z* is described by the Eq. (11) and *η* is the linear

If the loading force is negative (i.e., directed backwards, towards the heating laser beam, Eq. 14), it is not surprising that an initial depression observed in the foil surface is opposite to the heating laser beam. Similar results have been reported in the literature (Scruby 1987) for thicker metal samples where the thin plate approximation was not applicable. Maximum amplitudes and shapes of the observed depression vary for different metals and are defined

In the case of plasma formation, the situation becomes more complex. The amplitude of the driving force can be estimated using an expression similar to Eq.(13), whereas the time profile of generated stress pulse is the subject of experimental study (Krehl, Schwirzke et al. 1975). Laser plasmas formation and their interaction with the surface is very complex and multi-variable problem, which can only be solved in the framework of some model assumption (Mora 1982). This is why direct experimental studies of laser-driven surface vibrations should be an essential part of any acoustic wave related desorption phenomena.

irradiated spot and outward from the spot along the surface. Temperature evolution at any moment, *t>τ*, and for any position, *z*>0, proceeds according to the following equation (Prokhorov, Konov et al. 1990):

$$T(z,t) = \frac{2AI\_{\text{max}}\mathcal{I}^{1/2}}{\text{K}} \cdot [t^{1/2}ierfc(\frac{z}{2(\gamma t)^{1/2}}) - (t-\tau)^{1/2} \cdot ierfc(\frac{z}{2[\gamma(t-\tau)]^{1/2}})] \tag{11}$$

where *γ* is the thermal diffusivity of the metal, which can be expressed as γ ρ = Κ /*c* . The function *ierfc*(*x*) is given by

$$\text{ierfc}(\mathbf{x}) = \pi^{-1/2} \left\{ \exp(-\mathbf{x}^2) - \mathbf{x} (1 - \text{erf}(\mathbf{x})) \right\} \tag{12}$$

where <sup>2</sup> 0 <sup>2</sup> ( ) exp( ) = − *x erf x* ξ ξ*d* π

Eqs. (10) and (11) are the solutions of the one-dimensional heat diffusion equation and are valid only if the laser beam size, *r0,* is significantly greater than both the foil thickness *h* and the thermal diffusion length *lth* calculated as *lth*=(*γ*·*t*)1/2.

The strong rise of the surface temperature given by Eq.(10) results in the surface melting and evaporation, as well as in plasma plume formation (ablation regime) (Miller and Haglund 1998). Despite the fact that laser plasma generation and evolution have been the focus of numerous studies, no general mechanisms exist that describe the plasma recoil pressure on the surface for a broad range of laser intensities (Phipps, Turner et al. 1988), due to the complexity of the phenomenon. For GW/cm2 peak laser powers, hot and dense plasma is formed in the vicinity of the surface, which can screen the surface and prevent laser radiation from reaching it. In this case, the ablative pressure very weakly depends on the target material parameters (Phipps, Turner et al. 1988) and has a sub-linear dependence on laser intensity. In a semi-regulating, one-dimensional plasma model (which can be applied to our case as a simplified, first-order approximation), this equation is written, as follows (Gospodyn, Sardarli et al. 2002) :

$$P\_{a,\text{max}} = 7.26 \cdot 10^8 \cdot I^{3/4} \cdot (\mathcal{A} \cdot \sqrt{\pi})^{-1/4} \,, \tag{13}$$

where *I* is expressed in GW/cm2, *λ* in microns, *τ* in nanoseconds and *Pa,max* in Pa. While Eq. (13) was derived for an aluminum target in vacuum and for a supercritical plasma density, it exhibits only a weak dependence on the atomic mass, *A,* of the material irradiated (*A*-1/8) (Gospodyn, Sardarli et al. 2002) and may be applicable to specific experiments only as an upper limit estimate. For lower laser intensities (<1 GW/cm2), the plasma plume transmittance strongly varies with laser intensity (Song and Xu 1997), depending upon the plasma density and temperature. In this case the evaporated surface material is ionized only partially and the total mass of the evaporated atomic cloud are exponentially increasing with the surface temperature and, hence with the laser intensity (Murray and Wagner 1999).

### **2.2.2 Thermal and plasma driven acoustic waves in metal foils**

The temperature rise, as heat is transported into the solid causes linear thermal expansion resulting in the development of thermoelastic acoustic waves in the irradiated metal. Eqs. (10) and (11) can be used to determine the driving force which produces the waves. In

irradiated spot and outward from the spot along the surface. Temperature evolution at any moment, *t>τ*, and for any position, *z*>0, proceeds according to the following equation

> <sup>2</sup> (,) [ ( )( ) ( )] 2( ) 2[ ( )] = ⋅ −− ⋅ Κ − *AI <sup>z</sup> <sup>z</sup> Tzt t ierfc t ierfc t t*

> > 1/2 <sup>2</sup> ( ) {exp( ) (1 ( ))} <sup>−</sup> *ierfc x* = −− −

Eqs. (10) and (11) are the solutions of the one-dimensional heat diffusion equation and are valid only if the laser beam size, *r0,* is significantly greater than both the foil thickness *h* and

The strong rise of the surface temperature given by Eq.(10) results in the surface melting and evaporation, as well as in plasma plume formation (ablation regime) (Miller and Haglund 1998). Despite the fact that laser plasma generation and evolution have been the focus of numerous studies, no general mechanisms exist that describe the plasma recoil pressure on the surface for a broad range of laser intensities (Phipps, Turner et al. 1988), due to the complexity of the phenomenon. For GW/cm2 peak laser powers, hot and dense plasma is formed in the vicinity of the surface, which can screen the surface and prevent laser radiation from reaching it. In this case, the ablative pressure very weakly depends on the target material parameters (Phipps, Turner et al. 1988) and has a sub-linear dependence on laser intensity. In a semi-regulating, one-dimensional plasma model (which can be applied to our case as a simplified, first-order approximation), this equation is written, as follows

8 3/4 1/4

λ τ

,max 7.26 10 ( )<sup>−</sup> ≈ ⋅ ⋅ ⋅⋅ *P I <sup>a</sup>*

**2.2.2 Thermal and plasma driven acoustic waves in metal foils** 

where *I* is expressed in GW/cm2, *λ* in microns, *τ* in nanoseconds and *Pa,max* in Pa. While Eq. (13) was derived for an aluminum target in vacuum and for a supercritical plasma density, it exhibits only a weak dependence on the atomic mass, *A,* of the material irradiated (*A*-1/8) (Gospodyn, Sardarli et al. 2002) and may be applicable to specific experiments only as an upper limit estimate. For lower laser intensities (<1 GW/cm2), the plasma plume transmittance strongly varies with laser intensity (Song and Xu 1997), depending upon the plasma density and temperature. In this case the evaporated surface material is ionized only partially and the total mass of the evaporated atomic cloud are exponentially increasing with the surface temperature and, hence with the laser intensity (Murray and Wagner 1999).

The temperature rise, as heat is transported into the solid causes linear thermal expansion resulting in the development of thermoelastic acoustic waves in the irradiated metal. Eqs. (10) and (11) can be used to determine the driving force which produces the waves. In

1/2 1/2

*x x erf x* (12)

γ τ

γ

, (13)

(11)

 ρ= Κ /*c* .

τ

max 1/2 1/2

where *γ* is the thermal diffusivity of the metal, which can be expressed as

π

γ

(Prokhorov, Konov et al. 1990):

The function *ierfc*(*x*) is given by

where <sup>2</sup>

*erf x*

0 <sup>2</sup> ( ) exp( ) = − *x*

π

(Gospodyn, Sardarli et al. 2002) :

1/2

γ

ξ ξ*d*

the thermal diffusion length *lth* calculated as *lth*=(*γ*·*t*)1/2.

accordance with the general theory of thermal stresses in thin plates (Boley and Weiner 1960), a non-uniform heating of the surface is equivalent to a *negative loading pressure* and may be expressed as

$$P\_T = -\frac{N \cdot \eta}{1 - \varepsilon} \cdot \nabla\_r^2 \left[ \int\_{-h/2}^{h/2} T(r, z) \cdot z dz \right],\tag{14}$$

where the temperature distribution over *z* is described by the Eq. (11) and *η* is the linear thermal expansion coefficient.

If the loading force is negative (i.e., directed backwards, towards the heating laser beam, Eq. 14), it is not surprising that an initial depression observed in the foil surface is opposite to the heating laser beam. Similar results have been reported in the literature (Scruby 1987) for thicker metal samples where the thin plate approximation was not applicable. Maximum amplitudes and shapes of the observed depression vary for different metals and are defined by both the temperature profile (Eq. (11)) and by elastic properties of the material.

In the case of plasma formation, the situation becomes more complex. The amplitude of the driving force can be estimated using an expression similar to Eq.(13), whereas the time profile of generated stress pulse is the subject of experimental study (Krehl, Schwirzke et al. 1975). Laser plasmas formation and their interaction with the surface is very complex and multi-variable problem, which can only be solved in the framework of some model assumption (Mora 1982). This is why direct experimental studies of laser-driven surface vibrations should be an essential part of any acoustic wave related desorption phenomena.
