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P1: ZCK Final Pages Qu: 00, 00, 00, 00 Encyclopedia of Physical Science and Technology EN008C-361 June 29, 2001 13:58 Laser–Materials Interactions Michael Bass Aravinda Kar University of Central Florida I. Deposition of Light Energy in Matter II. Laser Heating of Materials III. Responses of Materials GLOSSARY arises from the fact that laser light can be much more intense than light from other sources. This property of laser Complex propagation constant Quantity γ (= α + jβ) light has opened new research areas in nonlinear optics, used to describe the propagation of light in a general laser plasma generation, and new types of materials pro- medium; α gives the attenuation and β the wave vector cessing. It has also led to applications such as laser cutting, magnitude, both in reciprocal centimeters. drilling, welding, and heat treating. The high fluxes possi- Laser-driven plasma Optically thick plasma found near ble allow consideration of such futuristic applications as a surface heated to the point of vaporization by an in- power generation through inertial confinement fusion, X- tense laser beam; the beam ionizes some of the vapor ray generation for microelectronics, and laser weaponry. and through the process of avalanche breakdown cre- ates the plasma. Nonlinear absorption Process in which the attenuation I. DEPOSITION OF LIGHT ENERGY coefficient itself is a function of the light intensity. IN MATTER Photoablation Process by which short-wave-length laser light breaks molecular bonds in matter and causes ma- A. Absorption terial to be ejected from the surface with very little The interaction between light, a form of electromagnetic heating of the sample. energy, and materials takes place between the optical elec- Surface electromagnetic wave Wave formed on an irra- tric field and the most nearly free electrons in the medium. diated surface when an irregularity on the surface scat- In quantum mechanical terms, one speaks of interactions ters the light energy; the amount of energy coupled can between photons and electrons in the presence of the lat- be significant. tice of nuclei. The lattice must be present to ensure momentum conservation. For the purposes of this article, the THE INTERACTION of laser light with matter is in many classical viewpoint of an electric field causing electrons ways like the interaction of ordinary light with matter. to move in the presence of damping or frictional forces is The difference between them and the interest in laser light sufficient to describe the process of optical absorption. 247 P1: ZCK Final Pages Encyclopedia of Physical Science and Technology EN008C-361 June 29, 2001 13:58 248 Laser–Materials Interactions Conduction band electrons in metals are able to inter- Note that if we had treated the lossy dielectric as a medium act with an optical electric field. Their motion is damped having a finite conductivity σ , we would have obtained by collisions with the vibrating lattice and so some of the 1/2 γ = jω{(µε)[1 − j(σ/ωε)]} , (8) light energy is transferred to the lattice. In this manner the material is heated. In semiconductors the motions of both where ε is the dielectric permittivity. It is clear that Eqs. (7) electrons in the conduction band and holes in the valence and (8) are identical with ε replaced by ε1 and σ/ω by −ε2. band must be considered. In dielectrics the electrons are Lossless dielectrics are media having σ identically zero. effectively bound to the atoms or molecules that compose The complex propagation constant is then purely imagi- the material. The applied optical field induces a polariza- nary and there is no absorption in the medium. tion in the material. Upon relaxation some of the energy in the polarization is coupled to the lattice and the material is b. Perfect metals. A perfect metal is a material hav- heated. These processes of absorption of energy from an ing an infinite conductivity or a material in which optical field can be treated by classical electromagnetics. α/ωε ≫ 1. That is, Maxwell’s equations, the constitutive equations of matter, and the boundary conditions for each material In this case we find can be solved in each case. 1/2 α = β = (ωµσ/2) , (9) which means that a field propagating in a metal will be 1. Absorption from an Electromagnetic attenuated by a factor of 1/e when it has traveled a distance Point of View 1/2 δ = (2/ωµσ) . (10) a. Propagation in a general dielectric material. Let the propagation of the optical electromagnetic wave in The quantity δ is called the skin depth and at optical fre- a medium be given by the complex propagation constant quencies for most metals it is ∼50 nm or about one-tenth γ where of the wavelength of green light. After a light beam has propagated one skin depth into a metal, its intensity is γ = α + jβ, (1) 2 reduced to 1/e or 0.135 of its value on the surface. The energy that is no longer in the electromagnetic wave so that a plane wave electric field is described by when light is attenuated in a metal drives conduction cur- E(z, t) = Re{E(0) exp(−γ z) exp[ j(ωt − βz)]}. (2) rents in the so-called skin layer. These in turn produce heating through ohmic losses and the light energy thereby Here ω is the radian frequency of the field, α the attenua- appears in the metal as heat. Heating of the metal at depths tion coefficient, and β the propagation coefficient. below the skin layer takes place by means of thermal A dielectric is described by the complex dielectric conduction. constant ε = ε1 + jε2, (3) c. Lossy, nonconductive dielectrics. As mentioned above, the results derived for a general dielectric are and a magnetic susceptibility µ. This enables us to define equivalent to those for a metal when the proper substitu- a complex index of refraction tions are made for ε1 and ε2. This allows us to consider the case where η = n + jk, (4) ε2/ε1 ≫ 1 in which n determines the propagation vector and k determines the attenuation of the wave in the medium. It is and write the attenuation coefficient for the lossy dielectric easy to show that as {[( ) ]/ } 2 2 1/2 1/2 1/2 n = ε 1 + ε2 + ε1 2 (5) α = ω(µε2/2) 1/2 and = (2π/λ0)(µRε2R/2) , (11) {[( 2 2)1/2 ]/ }1/2 where (µ0ε0)1/2 is recognized as the speed of light in free k = ε 1 + ε2 + ε1 2 . (6) space, ω = 2πc/λ0, and the subscript “R” indicates the By applying Maxwell’s equations and the dielectric relative permittivity and susceptibility. For electromag- boundary conditions as to the case of a lossy dielectric netic radiation with a free space wavelength of 1000 nm we find that propagating in such a dielectric we see that 1/2 4 −1 γ = jω{(µε1)[1 − j(ε2/ε1)]} . (7) α > 10 cm . P1: ZCK Final Pages Encyclopedia of Physical Science and Technology EN008C-361 June 29, 2001 13:58 Laser–Materials Interactions 249 This means that the electromagnetic energy will be re- 2 −4 duced to 1/e of its value within 10 cm. The nonconductive dielectric is a medium with very few free or conduction band electrons at room temperature and so we cannot consider ohmic losses as the mechanism for heating. Its interactions with electromagnetic radiation involve valence band electrons. When such electrons in this type of material absorb light energy they can be raised either to the conduction band or to some impurity state lying within the band gap. They then relax back to the valence band. The process of relaxation can be radiative FIGURE 1 Frequency dependences of ϵ1R and ϵ2R. (fluorescence and phosphorescence) or nonradiative. In either case phonons (lattice vibrations) are generated and the material is heated. The heat generation is localized to as the range of normal dispersion, and the region near to the region in which the absorption occurs and, as in ω = ω0 where it decreases with frequency is called the metals, heating beyond this region takes place by thermal range of anomalous dispersion. heat conduction. In the preceding discussion the contributions of the electronic polarizability to the dielectric constant were con- 2. Lorentz and Drude Models sidered. Other contributions occur such as stimulation of vibrations of ions in ionic crystals. This type of contri- The classical theory of absorption in dielectric materials bution is very small at optical frequencies because of the is due to H. A. Lorentz and in metals it is the result of the large mass of the ions compared to that of the electrons. work of P. K. L. Drude. Both models treat the optically In other words, the ionic polarizability is much smaller active electrons in a material as classical oscillators. In the than the electronic polarizability at optical frequencies. Lorentz model the electron is considered to be bound to We can therefore consider only the electronic terms when the nucleus by a harmonic restoring force. In this manner, evaluating optical absorption. By inserting Eq. (13) into Lorentz’s picture is that of the nonconductive dielectric. Eq. (11) we obtain the absorption coefficient of the lossy, Drude considered the electrons to be free and set the restor- nonconductive dielectric in terms of the properties of the ing force in the Lorentz model equal to zero. Both models electron’s kinematics assumed by Lorentz. That is, include a damping term in the electron’s equation of motion which in more modern terms is recognized as a result ( )1/2 ( ) Ŵω of electron–phonon collisions. 2 α = (2π/λ)(µR/2) Ne /ε0m ( ) . 2 These models solve for the electron’s motion in the ω2 − ω2 + Ŵ2ω2 0 presence of the electromagnetic field as a driving force. (14) From this, it is possible to write an expression for the The Drude model for metals assumes that the electrons polarization induced in the medium and from that to derive are free to move. This means that it is identical to the the dielectric constant. The Lorentz model for dielectrics Lorentz model except that ω0 is set equal to zero. The real gives the relative real and imaginary parts of the dielectric and imaginary parts of the metal’s dielectric constant are constant as then given by 2 2 ( ) ω − ω 2 0 ( ) 1 ε1R = 1 + Ne /ε0m ( 2 2)2 2 2 (12) ε1R = 1 − Ne2ε0m 2 2 (15) ω − ω + Ŵ ω ω + Ŵ 0 ( ) Ŵ and 2 ε2R = Ne ε0m . (16) 2 2 ( ) Ŵω ω(ω + Ŵ ) 2 ε2R = Ne /ε0m ( ) . (13) 2 ω2 − ω2 + Ŵ2ω2 The quantity Ŵ is related to the mean time between 0 electron collisions with lattice vibrations. T (i.e., to the In these expressions N is the number of dipoles per unit problem of electron–phonon scattering). By considering volume, e the electron charge, m the electron mass, Ŵ the the motion of electrons able to make collisions with lattice damping constant, ω0 the resonance radian frequency of vibrations in an electric field E having radian frequency the harmonically bound electron, ω the radian frequency ω, it is straightforward to show that the average velocity is of the field, and ε0 the permittivity of free space. Equa- eE T tions (12) and (13) are sketched in Fig. 1. The range of ν = − . (17) frequencies where ε1 increases with frequency is referred m 1 − jωT P1: ZCK Final Pages Encyclopedia of Physical Science and Technology EN008C-361 June 29, 2001 13:58 250 Laser–Materials Interactions The conductivity at this frequency is then applying to it the real and imaginary parts of the complex index of refraction for a metal–air interface, we can write 2 Ne T 1 σ = , (18) the field reflectivity. Multiplying this by its complex con- m 1 − jωT jugate, we find the intensity reflection coefficient for a where the dc conductivity is given by metal 2 σdc = Ne T/m. (19) RI = 1 − 2µε0ω/σ (24) From Eqs. (15) and (16) we see that if we allow Ŵ = 1/T, Since the conductivity σ decreases with increasing tem- then perature, R1 decreases with increasing temperature. As a ( ) result, more incident energy actually gets into the metal σdc 1 ε1 = 1 − T (20a) and is absorbed when the temperature is raised. This is 2 2 ε0 ω T + 1 true even though the absorption at high temperatures takes and place in a deeper skin depth. ( )( ) σdc 1 1 ε2 = . (20b) 2 2 ε0 ω ω T + 1 B. Nonlinear Absorption At electromagnetic field frequencies that are low, that Since the advent of lasers made possible highintensity op- is, when ωT ≪ 1, we have tical fields it has been possible to explore interactions of light with matter in which the response of the material ε1 = 1 − σdcT/ε0 (21) is not linear with the optical electric field. This subject, and known as nonlinear optics, has become a major field of research and has led to such useful optical devices as fre- ε2 = σdc/ε0ω (22) quency converters, tunable parametric devices, and opti- At such frequencies ε2 ≫ ε1, and since Ŵ = 1/T Eq. (14) cally bistable elements. This section presents a semiclas- gives sical discussion of nonlinear optics and how it gives rise 1/2 to nonlinear absorptions, in particular two-photon absorp- α = (ωµσ/2) , (23) tion (TPA). which is exactly the result we obtained earlier when treat- The polarization of a medium and the optical electric ing absorption from an electromagnetic point of view. In field applied to it are linked by the material’s susceptibility other words, the optical properties and the dc conductivity X, a tensor quantity. In the previous section we considered of a perfect metal are related through the fact that each is the limit of small optical fields, where the susceptibility is a determined by the motion of free electrons. At high fre- function of the dielectric constants only and is independent quencies transitions involving bound or valence band elec- of the field. In this case the polarization vector P is related trons are possible and there will be a noticeable deviation to the optical electric field E by the expression from this simple result of the Drude model. However, the P = X ∗ E (25) experimental data reported for most metals are in good agreement with the Drude prediction at wavelengths as Equation (25) is the relationship on which optics was built short as 1 µm. prior to 1961. As a result of lasers and the high fields they produce it is now necessary to allow that X can be a function of the optical field. This is accomplished by writing 3. Temperature Dependence the field-dependent susceptibility as a Taylor series expan- Another aspect of the absorption of light energy by met- sion in powers of the optical field and thus expressing the als that should be noted is the fact that it increases with polarization as temperature. This is important because during laser irra- P = X1 ∗ E + X2 ∗ E ∗ E + X3 ∗ E ∗ E ∗ E + · · · (26) diation the temperature of a metal will increase and so will the absorption. The coupling of energy into the metal Maxwell considered this form of relation in his classical is therefore dependent on the temperature dependence of treatise on electricity and magnetism but, for simplicity, the absorption. This property is easy to understand if we retained only the first-order term. remember that all the light that gets into a metal is ab- The first term on the right of Eq. (26) gives rise to sorbed in it. The question that must be addressed is how the “linear” optics discussed previously. The second term the amount of incident optical energy that is not reflected gives rise to optical second-harmonic generation or fre- from the metal’s surface depends on temperature. Recall- quency doubling and optical rectification. The third term ing the Fresnel expression for electric field reflectance and results in third-harmonic generation and self-focusing. P1: ZCK Final Pages Encyclopedia of Physical Science and Technology EN008C-361 June 29, 2001 13:58 Laser–Materials Interactions 251 Absorption processes involving one or more photons are also described by the susceptibilities Xn employed in Eq. (26). By considering the continuity equation for the flow of energy and the fact that the susceptibilities are complex quantities. ′ ′′ Xn = X n + jXn, the imaginary part of the third-order term can be shown to give rise to an altered form of Beer’s law. That is, I (0) exp(−α0z) I (z) = (27) 1 + (αTPA/α0)I (0)[1 − exp(−α0z)] where α0 = 4πkX1 is the conventional linear attenuation coefficient, k the propagation vector in the medium, and αTPA the nonlinear attenuation coefficient, given by 2 FIGURE 2 Plasma generated by a pulsed CO2 transverse elec- 32π ′′ αTPA = 2 ωX3 I (28) trical discharge at atmospheric pressure (TEA) laser incident from c the right on a copper target in vacuum. The outer region contains hot copper atoms. The central region is a very bright blue-white This is the part of the absorption that is linearly depen- color, indicating highly ionized, very hot copper vapor. dent on the intensity. From a quantum mechanical point of view, such a property is the result of processes in which two photons are absorbed simultaneously. In other words, a material with energy levels separated by U must be con- the irradiated region (see Fig. 2). This phenomenon occurs sidered able to absorb simultaneously two photons, each in almost all types of intense laser irradiation, continuous having energy U/2. As a result, when studying the absorp- wave (cw) or pulsed, on surfaces, and in the bulk of solids, tion of light in materials it is no longer sufficient to measure liquids, and gases. It can prevent the laser light from further it at one low intensity. Contributions from such higher or- coupling to the sample or it can enhance the coupling. der processes as TPA can occur and must be measured. Its most important applications are in driving the target For example, TPA is sufficient in some semiconductors compression used to produce fusion energy by inertial to enable their excitation to lasing inversions. It is also confinement and in creating point sources of soft X-rays a powerful tool for studying the presence of deep-level for X-ray lithography in microelectronics. dopants in semiconductors, since they contribute energy When an intense laser beam is focused inside a trans- levels that enhance TPA. parent medium a laser-induced break-down and plasma Higher-order multiphoton absorption processes are pos- can be formed in as short a time as 6 psec. This very fast sible. There is some evidence that three-photon pro- formation is the result of the very highly nonlinear processes have been detected. However, nonlinear processes cess of plasma formation. The process is so complex that higher than TPA require optical fields that are very high. it remains a major subject of current research. As a result, These fields may result in TPA, and the electrons that are only a qualitative discussion is presented in this article. thereby freed may be accelerated by the optical field to If the target reaches a high enough temperature to in- form a catastrophic electron avalanche breakdown. When duce mass loss, it is essential to account for the energy this happens the material’s effective absorptivity becomes carried away by the removed material in order to deter- 100%, too much energy is deposited in the irradiated vol- mine the energy absorbed. Mass can be removed by such ume, and severe mechanical damage known as intrinsic processes as vaporization, melt removal, and pyrolysis or laser-induced damage follows. This sequence of events is burning. When vaporization occurs some of the ejected responsible for setting intensity limits for materials and, material can be ionized. Then, through the process of in- consequently, for fixing the minimum size of optical com- verse bremsstrahlung, the vapor can be broken down by ponents used in high-power lasing systems. the optical field to produce an optically thick plasma. This plasma is a very hot blackbody radiator and emits an intense blue-white light called a laser-induced spark. C. Laser-Driven Plasma Coupling As most irradiations take place in air, we consider that and Decoupling case. At laser intensities slightly greater than the plasma One of the most spectacular features of laser–materials formation threshold intensity, a laser-supported combus- interactions is the formation of a laser-driven plasma in tion (LSC) wave can be ignited. These waves occur at P1: ZCK Final Pages Encyclopedia of Physical Science and Technology EN008C-361 June 29, 2001 13:58 252 Laser–Materials Interactions and the plasma properties are determined by its isentropic expansion. The LSD wave plasma expansion away from the surface is very rapid. As a result, though it intercepts and absorbs incoming laser energy, the LSD wave plasma does not reradiate this energy as strongly absorbed ultraviolet light into the area originally irradiated. Computations for two-dimensional LSD wave plasmas show that for short laser pulses (i.e., shorter than 1 µsec) FIGURE 3 Sketch of a one-dimensional laser-supported com- the overall coupling of light to metal targets can be as large bustion (LSC) wave. The hot plasma near the target can radiate as 25%. Since most metals have absorptions near 1–2% ultraviolet light, which is strongly absorbed by the target material. The LSC wave is of low enough density to allow the laser light to in the infrared, this represents a substantial enhancement reach and heat the plasma. of the coupling. However, the coupling remains the same for increasing intensities but is spread out over increasing 4 7 2 areas compared to that of the irradiated spot. Therefore, intensities from 2 × 10 to 10 W/cm with both pulsed for short pulses, while the total coupling coefficient may and cw lasers. The ignition of an LSC wave takes place in be large, the energy deposited in the irradiated area may the ejected target vapor and the heated vapor transfers en- be smaller than if the LSD wave plasma had never been ergy to the surrounding air. The LSC wave thus generated formed. propagates away from the target surface along the beam When an LSC wave plasma is formed, it is possible to path and drives a shock wave ahead of itself in the air (see find enhanced coupling in the irradiated region. This re- Fig. 3). sults from energy transfer by radiation from the hot, high- The nature of the coupling of the light to the material pressure plasma adjacent to the surface. The LSC wave when a plasma is formed depends on the beam parame- propagates into the air at a low speed, and a large fraction ters of energy, spot size, and pulse duration as well as air of the incoming laser energy is used to heat the plasma to pressure and target material. For long pulse times and low a temperature in excess of 20,000 K. This plasma radiates laser intensities the formation of an LSC wave usually re- very efficiently in the ultraviolet part of the spectrum, sults in decreased coupling as the wave propagates away where most materials, in particular, metals, absorb more from the surface. This is the case in most laser materi- strongly than in the visible and infrared. As long as the als processing applications requiring melting or material LSC wave expansion remains nearly one-dimensional removal (i.e., welding, drilling, and cutting). the plasma-radiated ultraviolet light remains localized At high intensities a laser-supported detonation (LSD) in the originally irradiated area and the local heating is wave is ignited. In this case absorption takes place in a enhanced. thin zone of hot, high-pressure air behind the detonation Efficient local coupling of incident energy to the sample wave (see Fig. 4). Since this wave takes air away from the requires that the plasma be ignited near the surface, an LSC surface, expansion fans form to satisfy the boundary con- wave be generated, and the plasma expansion remain one- ditions at the target surface. The plasma remains nearly dimensional during the irradiation. These requirements one-dimensional until the expansion fans from the edge define a range of intensities over which enhanced cou- reach the center. This time is given approximately by the pling will be observed. The minimum intensity is clearly beam radius divided by the speed of sound in the plasma. that which ignites a plasma at the surface in the vaporous In the vicinity of the surface there is no laser absorption ejecta. The maximum intensity is that which generates an LSD wave instead of an LSC wave. Pulse durations or dwell times of cw sources must be less than that required for radial expansion of the plasma to set in over the di- mension of the irradiated spot. Thus, enhanced coupling will occur for pulses shorter than the spot radius divided 5 by the speed of sound in the plasma (∼5 × 10 cm/sec). Once radial expansion begins, the pressure and the plasma temperature drop rapidly. Concomitantly, so does the ef- ficiency of the plasma as an ultraviolet light radiator. FIGURE 4 Sketch of a laser-supported detonation (LSD) wave. As mentioned above, the irradiation conditions used in The incoming laser light is absorbed in the LSD wave front as many laser materials processing applications correspond is propagates away from the target. This spreads any reradiated energy in the ultraviolet over a large area and reduces the local to the two-dimensional LSC wave plasma case. Therefore, energy deposition. plasma formation reduces the coupling of laser energy into P1: ZCK Final Pages Encyclopedia of Physical Science and Technology EN008C-361 June 29, 2001 13:58 Laser–Materials Interactions 253 the irradiated region, and further increases in the laser in- and t is the pulse duration. The diameter of the plume is tensity, either by increasing the laser power or decreasing considered to be linearly proportional to its height, that is, the spot size, are futile. To deal with this problem. He cover D = shp, where s is the slenderness ratio. In this equation, gas is often used to suppress plasma formation. In some the amount of material contained in the plume is equated cases strong gas jets are used to blow the plasma away and to the amount of material vaporizing at the liquid–vapor allow the laser light to enter the irradiated material. interface. Another aspect of laser-driven vaporization and plasma The energy balance for the plume temperature is given formation that should be considered is the momentum by the following equation. transferred to the surface. In leaving the surface at high Plume energy balance: speed the ejected material carries away a substantial −µhp amount of momentum. This appears as a recoil momen- P(1 − e ) − 2πrhpheff(T − T∞) = 0, tum of the surface and, since it occurs during the short where P, m, heff, T, and T∞ are the laser power, absorp- duration of the laser irradiation, it results in a substan- tion coefficient of the plume for the incident laser, effec- tial impulse to the surface. This process has been used tive heat transfer coefficient, absolute temperatures of the to shock-harden certain materials. An ablative coating is plume, and surrounding, respectively. The first term on placed on the area to be shock-hardened and it is struck the left-hand side of this equation represents the energy by a pulsed laser beam. The impulse due to the removal of absorption by the plume. This term is obtained from the the ablative coating produces the desired hardening. The Beer–Lambert law by considering the plume as the attenu- transfer of an impulse by laser irradiation is another sub- ating medium. The second term corresponds to the energy ject of major research interest, in part due to its potential loss from the plasma to the surrounding due to the radi- value as a laser weapons effect. ation and convection effects that are combined together by defining an effective heat transfer coefficient heff as 1. Analysis of Laser–Plasma Interaction follows: ( ) 2 2 A vapor–plasma plume consisting of ions, electrons, and heff = hc + εσ Tv + T∞ (Tv + T∞), vapor particles is generally formed above the workpiece where hc is the convection heat transfer coefficient, ε is the during various types of laser materials processing. The emissivity of the plume, s is the Stefan-Boltzmann con- laser beam propagates through this plume before reaching stant, and Tv is the boiling temperature of the workpiece. the substrate surface. It loses a considerable amount of its energy in the plume above a certain critical power density Equation of state for the plume: of the incident beam for a given process. Such partitioning pV = RTs, of laser energy between the plume and substrate is important because it determines the amount of laser energy that where p is the pressure exerted on the melt surface by the is actually utilized for materials processing. plume, R is the universal gas constant, Ts is the melt sur- Several physical phenomena need to be considered to face temperature, and V is the plume volume per unit mole. analyze the partitioning of laser energy during materi- Clasius–Clapeyron equation for vaporization: als processing. These include laser heating, melting, and vaporization, conservation of mass, momentum and en- ln(p/pc) = 4.693(1 − Tc/Ts), ergy, absorption and scattering of the laser radiation in where Pc and Tc are the critical pressure and tempera- the plume, gas dynamics, plasma dynamics, and equa- ture, respectively, which are given as Tc = 9340 K and tion of state. The following analysis provides a simpli- Pc = 10291.8 atm for iron. fied approach to understand this complex phenomenon of laser energy partitioning under quasi-steady-state condi- Plasma ionization (Saha-Eggert equation): tions for a pulsed laser beam, which is stationary relative 3/2 nIne 2U1(2πmekT) exp[−I/KT] to the substrate. = , 3 nneu U0h Mass balance: where nI, ne, and nneu are the ion, electron, and neu- 2 2 tral densities, respectively. U1 and U0 are the statistical πr hpρv − πr 0ρlSvτ = 0, weights of the singly ionized and neutral states respec- where r, hp, and ρv are the radius, height, and density of the tively. E is the first ionization potential, h is the Planck vapor-plasma plume, respectively. ro is the radius of the in- constant, k is the Boltzmann constant, and me is the coming laser beam, ρ1 is the density of workpiece at melt- mass of an electron. At high vapor temperatures, there ing temperature, Sv is the liquid-vapor interface velocity, is sufficient number of charged particles in the plume and P1: ZCK Final Pages Encyclopedia of Physical Science and Technology EN008C-361 June 29, 2001 13:58 254 Laser–Materials Interactions the electron-ion inverse bremsstrahlung process becomes dominant. The absorption coefficient for this process is given by the Bremsstrahlung absorption coefficient: 2 6 ni ne Z e ln[2.25KT/hv] 2 η = √ λ , 3 3 3 24π ε 0c menKT 2πmeKT where Z is the charge number, e is the electronic charge, c is the velocity of light in vacuum, and eo is the permittivity of free space. n and 1 are the frequency and wavelength of the laser light. Stefan condition for vaporization: [ ] Ts − Tm AI + kl = ρl Sν Lν (Sm − Sν)τ Stefan condition for melting: ( ) ( ) Ts − Tm Tm − T∞ kl − ks √ = ρ1LmSm, FIGURE 5 Partitioning of laser energy between the plasma and (Sm − Sv)τ 2 ατ workpiece. where A is the absorptivity of the melt surface, and I is laser energy that reaches the substrate surface after the the laser irradiance at the melt surface. kl and ρl are the beam propagates through the plume. The theory overes- thermal conductivity and density of the substrate at the timates the results compared to the experimental results melting temperature, respectively. Lv and Lm are the latent because the theory is based on the Bremsstrahlung absorp- heats of boiling and melting, respectively. a is the thermal tion of the laser radiation, whereas the beam may also be diffusivity, ks is the thermal conductivity, and Sm is the attenuated due to the scattering of the beam in the plume. solid–liquid interface velocity. The irradiance (I) reaching the substrate surface and the substrate surface temperature are estimated by the following expressions respectively. D. Surface Electromagnetic Waves Beer–Lambert law: Researchers studying such diverse topics as laser pro- −µhp cessing of semiconductors, laser-induced damage, laser I = I0e . materials processing, and laser-driven deposition pro- Substrate surface temperature: √ cesses have observed the formation of “ripples” in the ir- 4τ radiated region (see Fig. 6). Such structures on the surface Ts = AI , πρ1ceffkl of metals, dielectrics, and semiconductors can be divided into two categories: (1) resonant periodic structures (RPS) where I0 is the irradiance of the incident laser beam at and (2) nonresonant periodic structures (NRPS). RPS are the top surface of the plume, ceff is the effective heat ca- formed due to the interaction of the incident laser beam pacity that accounts for the latent heat of melting and with the surface electromagnetic waves (electromagnetic specific heat capacity of the workpiece cps, which is taken waves that propagate across the surface), and their peri- as ceff = cps + Lm/Tm. The vapor–liquid interface velocity ods are influenced by the wavelength, polarization, and attains a constant value at the quasi-steady state depend- incidence angle of the beam. The RPS formation theory is ing on the plasma volume and temperature. The mass of generally based on (1) electrodynamic analysis, in which material vaporized per unit area per unit time (J ) is given the modulation of the energy absorbed at the substrate sur- by Vapor flux: face is studied by considering the interference between the J = Sνρl . incident and surface electromagnetic waves, and (2) ther- These equations describe various aspects of the pro- mophysical analysis, in which the nonuniform dissipation cess physics and represented a set of coupled nonlinear of energy at the substrate surface is studied by considering algebraic equations that need to be solved numerically. microscopic changes in the crystal structure during laser Typical results for the partitioning of laser energy is pre- heating. The formation of NRPS is generally attributed to sented in Figure 5 for stainless steel (SS 316) substrate the melting of metal surface in pulsed laser heating. It in- irradiated with a Nd:YAG laser of pulse duration 160 ns. volves nonuniform vaporization of the metal, and capillary Pinhole experiments were conducted to obtain the exper- oscillations of the molten surface layer due to instabilities imental results of this figure by actually measuring the at the plasma–substrate interface. P1: ZCK Final Pages Encyclopedia of Physical Science and Technology EN008C-361 June 29, 2001 13:58 Laser–Materials Interactions 255 tromagnetic theory of gratings to understand the process. From this theory, it is clear that for a particular line spacing, there is a particular grating height that can dramatically enhance the absorption of a material over that of a smooth surface. The generation of SEW can therefore result in greatly enhanced coupling of laser light to materials. On the other hand, if the surface waves propagate, they may carry energy away from the irradiated area. If the di- mension of the irradiated area is small compared to the SEW attenuation length, this may be a major factor in re- distributing the absorbed energy. The irradiated region will not become as hot as it might if no SEW were generated. It should be noted that this effect would give rise to a beam spot size dependence of laser heating when the beam diam- eters used are in the range of one SEW attenuation length. FIGURE 6 Scanning electron micrograph of ripples spaced by ∼10 µm produced in 304 stainless steel exposed to a cw CO2 E. Photoablation laser able to melt metal. The sample was stationary under a nor- A new and exciting light–matter interaction called pho- mally incident 10.6-µm beam polarized perpendicular to the ripples. The residual ripples are found near the edges of the melt toablation has been demonstrated with the advent of high pool, where freezing is sufficiently rapid to preserve the ripples. peak intensity ultraviolet lasers. This occurs with the In the central part of the melt pool convective motion in the melt excimer lasers (i.e., ArF, KrF, XeF, or XeCl) operating obscures the ripples. The ripples are the result of surface elec- at wavelengths from ∼250 to 350 nm. The lasers pro- tromagnetic waves (SEW) launched by scattering the incident vide high energies (∼1 J or more per pulse) in pulses of light and subsequent interference between two electromagnetic waves. ∼30 nsec duration. Most organic materials absorb very strongly at these wavelengths. The energies of photons in the ultraviolet are sufficient to efficiently break bonds in All of these observations had in common the facts that such materials. When this happens the fragments are sub- linearly polarized lasers were used and that the material jected to very high pressures and are ejected from the sur- in the irradiated region had reached its melting point. The face (see Fig. 7). The process takes place entirely within ripples were spaced by the light wavelength when the light one absorption depth (∼20 nm) and so provides a very was at normal incidence. These facts and additional data controllable means of material removal. In addition, the demonstrated that the ripples were the result of the following sequence of events: 1. The incident light was scattered by some surface irregularity and a surface electromagnetic wave was launched. 2. In the region irradiated this wave was able to interfere with the incident light. 3. The interference pattern produced localized differ- ences in the surface temperature, and when the material became molten it flowed due to surface tension effects or differential vapor pressures to conform with the varying light intensity. 4. The molten material froze quickly enough following irradiation that the ripples were detectable afterward. The key element in the formation of these ripples is the fact that surface electromagnetic waves (SEW) were generated. This is another means by which light energy can be coupled into a material and one that was not obvi- ous from prelaser knowledge. The fact that a gratinglike FIGURE 7 Sketch of the process of ultraviolet laser irradiation- interference pattern is set up allows one to apply the elec- induced photoablation. P1: ZCK Final Pages Encyclopedia of Physical Science and Technology EN008C-361 June 29, 2001 13:58 256 Laser–Materials Interactions energy deposited in the material is removed as kinetic energy of the ejecta. As a result, there is very little thermal heating of the sample and the process is confined to the area irradiated. With lasers operating at reasonably high repetition rates, this process can be used to remove significant amounts of material. The use of excimer lasers and photoablation is still under consideration in research and development laborato- ries. However, it is clearly suitable for removing protective coatings or insulation and for certain medical applications where great delicacy is required. II. LASER HEATING OF MATERIALS A. Thermal Diffusion Problem and Thermal Diffusivity We are concerned with the heating of materials that results from several of the laser light absorption processes discussed in the previous sections. This can be approached FIGURE 8 Sketch of coordinate system used in evaluating responses of materials to laser irradiation. in a number of nearly equivalent ways, all of which require a knowledge of (1) the target’s optical and thermal properties during the irradiation, (2) the laser beam dis- properties that will depend on temperature and position. tribution, (3) the dynamics of the irradiation process, and The quantity A(x, y, z, t) is the rate at which heat is (4) the processes of phase change in the target material. As supplied to the solid per unit time per unit volume in joules would be expected, it is extremely difficult to solve the heat per second per cubic centimeter and T = T (x, y, z, t) flow problem exactly in the general case and so reasonable is the resulting temperature distribution in the material. approximations are used. In addition, only problems that Figure 8 defines the coordinate system used. are easy to solve are attempted. These are then useful as The temperature dependence of the properties results guides to the solution of other problems. in a nonlinear equation that is very difficult to solve ex- Consider the equation of heat conduction in a solid with actly. Where the functional dependence of these quantities the laser energy absorbed on the irradiated surface as a on temperature is known, it is sometimes possible to use heat source. By judiciously selecting the beam geometry, numerical integration techniques to obtain a solution. A dwell time, and sample configuration, the problem may be further complication arises from the temperature depen- reduced to solvable one- and two-dimensional heat flow dence of A(x, y, z, t) through that of the material’s ab- analyses. Phase transitions can be included and the tem- sorptivity. When phase transitions occur one can attempt perature distributions that are produced can be calculated. a solution of the problem by solving for each phase sep- Examples will be selected to provide specific guidance in arately and including the heat required for the transition the choice of lasers and materials. The result of all this where appropriate. will be an idea of the effects that one may produce by For most materials the thermal properties do not vary laser heating of solids. greatly with temperature and can be assigned an average The equation for heat flow in a three-dimensional solid value for the temperature range to be studied. In this case is it is possible to solve the heat flow problem. A further sim- ( ) ( ) plification is obtained by assuming that the material is ho- ∂T ∂ ∂T ∂ ∂T ρC = K + K mogeneous and isotropic. Under these conditions Eq. (29) ∂t ∂x ∂x ∂y ∂y reduces to ( ) ∂ ∂T 1 ∂T A(x, y, z, t) 2 + K + A(x, y, z, t), (29) ∇ T − = − , (30) ∂z ∂z κ ∂t K where κ = K/ρC is the thermal diffusivity. In the steady where ρ is the material density in grams per cubic cen- state ∂T/∂t = 0, resulting in timeter, K the thermal conductivity in watts per centimeter per degree Celsius, and C the heat capacity in joules 2 A(x, y, z, t) ∇ T = − . (31) per centimeter per degree Celsius; these are material K