Laser Pulse Phenomena and Applications Part 3 doc

The Effect of the Time Structure of Laser Pulse on Temperature Distribution and Thermal Stresses in Homogeneous Body with Coating 51 For greater pulse rise times τr the dimensionless ti

The Effect of the Time Structure of Laser Pulse on Temperature Distribution

and Thermal Stresses in Homogeneous Body with Coating 51

For greater pulse rise times τr the dimensionless time τc, connected with the change of

stresses type from compressive to tensile one also increases (Fig 4) This dependence is

approximately described by the equation τc=0.8173τr2+0.075τr+0.1457

In thermal processing of brittle materials, it is the sign change of superficial stresses what

plays key role in controlled thermal splitting (Dostanko et al., 2002) The beginning of

superficial cracks generation is accompanied with the monotonic increase of tensile lateral

stresses and makes the controlled evolution of the crevices possible By equating the relation

(45) to 0 at τ τ= c>τs, 0ζ = and taking into account the equations (42), (48), (49), one

where the function Q(0)0 ( )τ has the form (66) With the absolute inaccuracy, smaller than 3%,

the solution of nonlinear equation (70) can be approximated by the function

0.6

τc

τsFig 5 Dimensionless time τc of the sign change of lateral stress σy∗ on the irradiated

surface 0ζ = versus dimensionless duration τs of the rectangular laser pulse (Yevtushenko

et al., 2007)

The forced cooling of the surface in the moment time t t= would cause the jump of c

temperature Δ =T T(0,t c− −0) T(0,t c+0) in thin superficial layer From the equations (45)

and (46) it follows, that the dimensionless lateral deformation of the plate εy∗ is determined

by the integral characteristic of temperature only For this reason, the rapid cooling of the

thin film, practically, does not change the surface deformation (0,εy t c+0)=αc T(0,t c−0) but

at the same moment it produces the increase of the normal stresses σy(0,t c+0)= Δ αc T

Finally, the development of the superficial crack can be described as a series of the following phases:

1 due to local short heating a surface of the sample in it the field of normal lateral stresses

on the processed surface and considerably higher than for the homogeneous half-space (one order of magnitude) lateral tensile stresses generated in the superficial layer when the heating is finished So, the thermal processing of the coating from zirconium dioxide leads

to the generation of superficial cracks, which divide the surface into smaller fragments Of course the distribution of cracks at different depths depends on the heat flux intensity, the diameter of the laser beam, pulse duration and other parameters of the laser system

But when using dimensionless variables and parameters the results can be compared and the conclusion is that for the heating duration τs=0.15, penetration depth of cracks for coating–substrate system (ZrO2–40H steel) is, more than two times greater than for the homogeneous material (one can compare Figs 3а and 6)

The opposite, to the discussed above, combination of thermo-physical properties of the coating and the substrate is represented by the copper–granite system, often used in ornaments decorating interiors of the buildings like theatres and churches For the copper coating K = c 402 W/(m K), k c=125 10 m /s⋅ −6 2 , while for the granite substrate 1.4 W/(m K)

s

K = , k s=0.505 10 m /s⋅ −6 2 , what means that the substrate is practically thermal insulator and the coating has good thermal conductivity (see Figs 7 and 9) The distribution of lateral thermal stresses for copper–granite system is presented in Fig 8 In this situation, when the thickness of the coating increases, the temperature on the copper surface decreases Therefore the effective depth of heat penetration into the coating is greater for the better conducting copper than for thermally insulating zirconium dioxide (ZrO2) (see Figs 6 and 8) We note that near to the heated surface ζ = lateral stresses 0 σyare compressive not only in the heating phase 0< <τ 0.15 but also during relaxation time, when the heat source is off Considerable lateral tensile stresses occur during the cooling phase close to the interface of the substrate and the coating, ζ = This region of the tensile 1stresses on the copper-granite interface can destroy their contact and in effect the copper coating exfoliation can result

The Effect of the Time Structure of Laser Pulse on Temperature Distribution

and Thermal Stresses in Homogeneous Body with Coating 53

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Fig 6 Isolines of dimensionless lateral stress σy∗ for ZrO2 ceramic coating and 40H steel substrate at rectangular laser pulse duration τs=0.15 (Yevtushenko et al., 2007)

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Fig 7 Isotherms of dimensionless temperature T∗ for ZrO2 ceramic coating and 40H steel substrate at rectangular laser pulse duration τs=0.15(Yevtushenko et al., 2007)

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Fig 8 Isolines of dimensionless lateral stress σy∗ for copper coating and granite substrate at rectangular laser pulse duration τs=0.15(Yevtushenko et al., 2007)

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Fig 9 Isotherms of dimensionless temperature T∗ for copper coating and granite substrate

at rectangular laser pulse duration τs=0.15(Yevtushenko et al., 2007)

The Effect of the Time Structure of Laser Pulse on Temperature Distribution

and Thermal Stresses in Homogeneous Body with Coating 55

6 Effective absorption coefficient during laser irradiation

The effective absorption coefficient A in the formula (1) and (12) is defined as the ratio of

laser irradiation energy absorbed on the metal’s surface and the energy of the incident beam

(Rozniakowski, 2001) This dimensionless parameter applies to the absorption on the metal’s

surface, on the very sample surface (so called “skin effect”) The absorption coefficient A

can be found in book (Sala, 1986) or obtained on the basis of calorimetric measurements

(Ujihara, 1972) The mixed method of effective absorption coefficient determination for some

metals and alloys was presented by Yevtushenko et al., 2005 This method is based on the

solution of axisymmetric boundary-value heat conduction problem for semi-space with

circular shape line of division in the boundary conditions and on the metallographic

measurements of dimensions of laser induced structural changes in metals The calculations

in this method are very complex because, in particular, the numerical calculation of the

Hankel’s integrals has to be done Therefore, we shall try to use with this purpose obtained

above the analytical solution of the transient one-dimensional heat conduction problem for

homogeneous semi-space in the form

0

where, taking the formula (12) into account, the coefficient T0′ =T0/A and the

dimensionless temperature T∗( , )ζ τ is defined by formulae (41) and (60) It should be

noticed that the temperature on the irradiated surface has maximum value at the moment of

laser switching off, for t t= (s τ τ= ), and in the superficial layers the maximum is reached s

for t t≡ h= + Δ (in dimensionless units, for t s t τh=τs+ Δ , τ Δ =τ k t / dΔ 2, d is the radius of

the irradiated zone) The parameter tΔ (Δ ) is known as “the retardation time” τ

(Rozniakowska & Yevtushenko, 2005) The time interval, when the temperature T reaches

its maximum in the point z z= h beneath the heated surface, can be found from the

condition:

( , )0

h

T z t t

From the equation (74) for the known dimensionless hardened layer depth ζh and the pulse

duration τs, the dimensionless retardation time Δ can be found On the other hand, at τ

known Δ from equation (74) we find the dimensionless hardened layer depth τ ζh of

maximum temperature can be found:

0 0.02 0.04 0.06 0.08 0.1 0

0.0002 0.0004 0.0006 0.0008

0.01

Δτ

ζhcobalt: τs=0.672

(b) Fig 10 Dimensionless hardened layer depth ζh from the heated surface versus dimensionless retardation time Δ , for the dimensionless laser pulse duration: а) τ τs=0.0732 (St.45 steel sample); b) τs=0.672 (Co monocrystal sample) (Yevtushenko et al., 2005)

The Effect of the Time Structure of Laser Pulse on Temperature Distribution

and Thermal Stresses in Homogeneous Body with Coating 57

parameter

metal

laser type mmd s

Table 1 Input data needed for the calculations of the effective absorption coefficients for the

St.45 steel and Co monocrystal samples

h

s

ττ

τ = and τs=0.672 are shown in Figs 10a and 10b, respectively The dimensionless

retardation time quickly increases with the distance increase from the heated surface The

dimensionless pulse durations τs were calculated from equation (12) with the use of

material constants characteristic for St 45 steel and Co monocrystals (Table 1), which were

presented by Yevtushenko et al., 2005

Assuming the temperature T h of the structural phase transition, characteristic for the

material, is achieved to a depth z h from the heated surface at the moment t h It should be

noticed that for steel the region of structural phase transitions is just the hardened layer,

while for cobalt – it is the region where, as a result of laser irradiation, no open domains of

Kittel’s type are observed

It can be assumed that the thickness of these layers z h, is known – it can be found in the

way described by Rozniakowski, 2001 Then, from the condition

where dimensionless temperature T∗ is expressed by equations (41) and (60), the

dimensionless retardation time Δ can be found from the equation (75) and the constant τ

T′ =q d K The input data needed for the calculations by formula (77) are included in

Table 1 Experimental results obtained by Rozniakowski, 1991, 2001 as well as the

solutions for the axisymmetric (Yevtushenko et al., 2005) and one-dimensional model are

presented in

Values of the effective absorption coefficient for the St.45 steel sample irradiated with pulses

of short duration τs=0.0732, found on the basis of the solutions for axisymmetric (A =0.42) and one-dimensional (A =0.41) transient heat conduction problem are nearly the same, and correspond to the middle of the experimentally obtained values range 0.3 0.5

A = ÷ (Table 2) The cobalt monocrystal samples were irradiated with pulses of much longer duration τs=0.672 In this case, there is more than twofold difference of A values

found on the basis of the solutions for axisymmetric (A =0.112) and one-dimensional (A =0.045) transient heat conduction problem Moreover, only the value of effective absorption coefficient obtained from the axisymmetric solution of transient heat conduction problem corresponds to the experimental value A =0.1 In that manner, it was proved that the solution of one-dimensional boundary heat conduction problem of parabolic type for the semi-space can be successfully applied in calculations of the effective absorption coefficient only for laser pulses of dimensionless short duration τs<< Otherwise, the solution of 1axisymmetric heat conduction problem must be used

7 Conclusions

The analytical solution of transient boundary-value heat conduction problem of parabolic type was obtained for the non-homogeneous body consisting of bulk substrate and a thin coating of different material deposited on its surface The heating of the outer surface of this coating was realised with laser pulses of the rectangular or triangular time structure

The dependence of temperature distribution in such body on the time parameters of the pulses was examined It was proved that the most effective, from the point of view of the minimal energy losses in reaching the maximal temperature, is irradiation by pulses of the triangular form with flat forward and abrupt back front

Analysis of the evolution of stresses in the homogeneous plate proves that when it is heated, considerable lateral compressive stresses occur near the outer surface The value of this stresses decreases when the heating is stopped and after some time the sign changes – what means that the tensile stresses takes place The time when it happens increases monotonously with increase of a thermal pulse duration (for rectangular laser pulses) or with increase of rise time (for triangular laser pulses) When the lateral tensile stresses exceed the strength of the material then a crack on the surface can arise The region of lateral compressive stresses, which occur beneath the surface, limits their development into the material

The Effect of the Time Structure of Laser Pulse on Temperature Distribution

and Thermal Stresses in Homogeneous Body with Coating 59 The presence of the coating (for example, ZrO2) with thermal conductivity lower than for the substrate results in considerably higher than for the homogeneous material, lateral tensile stresses in the subsurface after the termination of heating The depth of thermal splitting is also increased in this case When the material of the coating (for example, copper) has greater conductivity than the substrate (granite) then the stresses have compressive character all the time The coating of this kind can protect from thermal splitting The region that is vulnerable for damage in this case is close to the interface of the substrate and the coating, where considerable tensile stresses occur during the cooling phase

The method for calculation of the effective absorption coefficient during high-power laser irradiation based on the solution of one-dimensional boundary problem of heat conduction for semi-space, when heating is realised with short pulses, was proposed, too

8 References

Coutouly, J F et al (1999) Laser diode processing for reducing core-loss of gain-oriented

silicon steels, Lasers Eng., Vol 8, pp 145-157

Dostanko, A P et al (2002) Technology and technique of precise laser modification of solid-state

structures, Tiechnoprint, Minsk (in Russian)

Duley, W W (1976) CO 2 Lasers: Effects and Applications, Acad Press, New York

Gureev, D М (1983) Influence of laser pulse shape on hardened coating depth, Kvant

Elektron (in Russian), Vol 13, No 8, pp 1716-1718

Hector, L G & Hetnarski, R B (1996) Thermal stresses in materials due to laser heating, In:

R B Hetnarski, Thermal Stresses IV, pp 1-79, North–Holland

Kim, W S et al (1997) Thermoelastic stresses in a bonded coating due to repetitively

pulsed laser radiation, Acta Mech., Vol 125, pp 107-128

Li, J et al (1997) Decreasing the core loss of grain-oriented silicon steel by laser processing,

J Mater Process Tech., Vol 69, pp 180-185

Loze, M K & Wright, C D (1997) Temperature distributions in a semi-infinite and

finite-thickness media as a result of absorption of laser light, Appl Opt., Vol 36, pp

494-507

Luikov, A V (1986) Analytical Heat Diffusion Theory, Academic Press, New York

Ready, J F (1971) Effects of high-power laser radiation, Academic Press, New York

Rozniakowska, М & Yevtushenko, A A (2005) Influence of laser pulse shape both on

temperature profile and hardened coating depth, Heat Mass Trans., Vol 42, pp

64-70

Rozniakowski, K (1991) Laser-excited magnetic change in cobalt monocrystal, J Materials

Science, Vol 26, pp 5811-5814

Rozniakowski, K (2001) Application of laser radiation for examination and modification of

building materials properties, (in Polish), BIGRAF, Warsaw

Rykalin, N N et al (1985) Laser and electron-radiation processing of materials, (in Russian),

Mashinostroenie, Moscow

Said-Galiyev, E E & Nikitin, L N (1993) Possibilities of Modifying the Surface of

Polymeric Composites by Laser Irradiation, Mech Comp Mater., Vol 29, pp

259-266

Sala, A (1986) Radiant properties of materials, Elsevier, Amsterdam

Sheng, P & Chryssolouris, G (1995) Theoretical Model of Laser Grooving for Composite

Materials, J Comp Mater., Vol 29, pp 96-112

Timoshenko, S P & Goodier, J N (1951) Тheory of Elasticity, McGraw-Hill, New York Ujihara, K (1972) Reflectivity of metals at high temperatures, J Appl Phys., Vol 43,

pp 2376-2383

Welch, A J & Van Gemert, M J C (1995) Optical-thermal response of laser-irradiated tissue,

Plenum Press, New York and London

Yevtushenko, A A et al (2005) Evaluation of effective absorption coefficient during laser

irradiation using of metals martensite transformation, Heat Mass Trans., Vol 41,

pp 338-346

Yevtushenko, A.A et al (2007) Laser-induced thermal splitting in homogeneous body with

coating, Numerical Heat Transfer, Part A., Vol 52, pp 357-375

Fig 1 Ms Röntgen's hand First medical imaging with X-rays (December 22, 1985; source: wikipedia.org)

range of femtoseconds (1 fs = 10-15 s) is applied to the gas, a plateau of equally intense harmonics of very high order can be observed The atom is ionized when the absolute electric field of the laser is close to its crest during an optical cycle and is pulled away from the parent ion Since the laser electric field changes its sign about a quarter of a period later, the electron will slow down, stop at a position far from the ion and start to accelerate back towards it (Corkum, 1993) When it returns to the ion, it can possess significant amount of kinetic energy, much larger than the photon energy but being its multiple This energy plus the ionization potential is transferred into emitted photon energy as soon as the electron recombines with its parent ion, which gives rise to very high harmonic orders observed in the experiments (Macklin, 1993) Thus HHG represents a source of coherent X-rays bursts of ultrashort time duration Additionally, the HHG source features spectral tunability from UV

to hard X-rays Moreover, advantage of particular importance is a very high repetition rate

of HHG which is given by the repetition rate of the driving laser only and can be easily as much as few kHz (Schultze et al., 2007)!

It has been shown that high-order harmonic pulse comprises train of attosecond pulses (Papadogiannis et al., 1999) This great advantage constitutes a stimulus for further development of high-order harmonic sources, especially of the techniques leading to generation of single attosecond pulses Nowadays, well explored and most frequently deployed are:

• usage of very short IR laser pulses ( < 5 fs) (Christov et al., 1997; Baltuska et al., 2003),

• a technique called polarization gating (Sola et al., 2006)

The details of the aforementioned techniques will not be discussed in detail here; however,

it is worth noting that the intension of improvement of high-order harmonic sources has become a boost for laser technology progress leading to development of laser systems emitting pulses with duration in the range of single optical cycle (~ 3.3 fs at ~810 nm central wavelength) and shifting the laser pulse central wavelength to the mid-infrared spectral range (MIR) in around 2-3 μm Besides, the lasers’ repetition rates have been significantly increased typically to a few kHz (and energy ~mJ per pulse; e.g Schultze et al., 2007) Another recent achievement of particular interest is carrier-envelope absolute phase stabilization (CEP)

State-of-the-art HHG sources require not only development of the high-harmonic source itself but also sophisticated metrology techniques and methods for characterization of femtosecond and attosecond pulses (Véniard et al., 1996; Drescher et al., 2002; Kienberger et al., 2002; Mairesse et al., 2005; Itatani et al., 2002; Sansone et al., 2008)

Due to unusual combination of all properties that high-order harmonics feature, they immediately found vast number of unprecedented applications For example, a number of

Fig 2 Typical spectrum of high-order harmonics (conversion medium: argon; Jakubczak a))

High-order Harmonic Generation 63 experimental results have been recently published related to time-resolved investigation of atomic processes For instance manipulation of drift energy of photoelectron wave packets (so called "steering of wave packets") and their imaging (e.g Kienberger et al., 2007), measurement of relaxation and lifetime dynamics in an atom by the direct measurement in time domain with attosecond resolution (e.g Baltuska et al., 2003; Kienberger et al., 2002) in contrary to thus far frequency-domain measurements of transition linewidths (Becker & Shirley, 1996), spectroscopy of bound electron dynamics in atoms and molecules (Hentschel

et al., 2001), observation of interference of coherent electron wave packets (Remetter et al., 2006), probing molecular dynamics (Niikura et al., 2002) and real-time tomography of molecular orbitals (Itatani et al., 2004)

Moreover, novel and very promising schemes for HHG have been recently demonstrated, e.g., generation of harmonics during reflection of super intense ultrashort IR laser pulses (I > 1017 W/cm2) from plasma mirror oscillating at relativistic velocities on the surface of a solid state target (Quéré et al., 2006), or generation of HHG from interaction of IR femtosecond laser pulses with molecules (N2, H2+; Lorin et al., 2008)

2 Physical mechanisms of high-order harmonic generation

If material is subjected to a strong electric field, nonlinear polarization of the material is induced The magnitude of the arisen polarization strongly depends on the intensity of the incident radiation At moderate and low intensity values the external electric field does not influence significantly the electronic structure of the irradiated atoms The potential barriers can be just slightly modified and Stark effect can be observed To great probability the atoms remain in their ground state and extension of their ground state wave function is of the order of Bohr radius (5.2917 10 m⋅ -11 ) All nonlinear phenomena taking place in this regime

are well described by the perturbation theory Thus it is referred as the perturbative regime of

nonlinear optics Comprehensive discussion on phenomena and related theory in the perturbative regime can be found e.g in Boyd, 2003 Some of nonlinear optical phenomena

in this regime are:

• harmonic radiation generation (second, third, etc.),

• optical parametric amplification,

Range of intensities implying these phenomena defines the strong field nonlinear optics regime

In contrary to the perturbative regime, here, the nonlinear response of the polarization of the medium is affected by the ionization process The nonlinear treatment can be only applied

Fig 3 Tunnel ionization The atomic potential affected by the external electric field whose the field strength is comparable to the atomic fields It is plausible that the electrons from the most-outer atomic will be unbound This transition is often referred as optical field

ionization (OFI)

Fig 4 In this case, the applied electric field is higher than the atomic field strength The atomic potential barrier is suppressed and electrons from most-outer shells are liberated through above barrier ionization

to an electron which is in very close vicinity of a parent ion As soon as it is released by optical field its response is linear to the electric field and may be treated by classical laws of motion (Corkum, 1989; Corkum, 1993)

Very interesting phenomena are present in the intermediate range of parameters, in the so

called intermediate regime, i.e between the perturbative and the strong field regimes They

include long-distance self-channeling when nonlinear Kerr effect causes beam focusing, on the one hand, and free electrons cause its defocusing, on the other This interplay leads to the channeling of the propagating intense pulse (even at distances as long as a few meters) Another interesting phenomenon in this regime is multiphoton ionization, where the total amount of absorbed energy exceeds the ionization potential (Fig 5)

When electric field strengths are even higher, the nonlinearities become stronger Electric field is able to optically liberate electrons from inner shells of the atom and the wiggling energy of an electron is comparable with its rest energy mc2 This is a launch of relativistic regime

Publications of crucial importance related to the intermediate to strong-field nonlinear optics regimes were made by Keldysh (Keldysh, 1965) and Ammonsov, Delone and Krainov (Ammosov et al., 1986) Keldysh defined a parameter, which was later named after him that allows determining whether tunneling or multiphoton process is dominant for particular

experimental conditions It reads:

High-order Harmonic Generation 65

Fig 5 Multiphoton ionization process: n-photons are absorbed The total energy of absorbed

photons (n*hν; n - number of absorbed photons, h - Planck's constant, ν - light frequency)

exceeds ionization potential

2

p p

I U

γ=

Where:

Ip - is ionization potential of a nonlinear medium,

Up - is ponderomotive potential, which is cycle-averaged quivering energy of an electron in

the external laser field It is defined as:

0 2

4

p e

e E U

E0 - external field amplitude oscillating at frequency ω

Substitution of constants leads to simplified relation: