Laser beam interaction with materials for microscale applications A Dissertation submitted to the Faculty of the Worcester Polytechnic Institute in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Mechanical Engineering by Krzysztof A. Nowakowski 22 November 2005 Approved: ___________________________________________________________ Prof. Ryszard J. Pryputniewicz, Major Advisor ____________________________________________________________________ Prof. Cosme Furlong, Member, Dissertation Committee ____________________________________________________________________________ Prof. Richard D. Sisson Jr., Member, Dissertation Committee Dr. Thomas F. Marinis, Draper Laboratory, Cambridge, MA Member, Dissertation Committee Prof. John M. Sullivan, Jr., Graduate Committee Representative This Dissertation is dedicated to my Wife and Children 2 Copyright © 2005 by NEST – NanoEngineering, Science, and Technology CHSLT – Center for Holographic Studies and Laser micro-mechaTronics Mechanical Engineering Department Worcester Polytechnic Institute Worcester, MA 01609-2280 All rights reserved 3 Nomenclature a is spot size if the collimated beam exiting the laser ai coefficients of the least-square fit function b the focused beam size of the lease beam on the fiber face c specific heat, or speed of light, or the desired fiber fill factor c(T) specific heat cp specific heat at constant pressure cpl specific heat at constant pressure for liquid state cpv specific heat at constant pressure for vapor state cs velocity of sound in the solid d minimum fiber optic diameter dA differential area dmin size of the minimum spot dt the infinitesimal time step dx spatial increments in x direction dy spatial increments in y direction dz spatial increments in z direction ecc eccentricity erf error function erfc complementary error function f laser pulse frequency, or focal length, or fraction of electron energy fcf focusing correction factor fs femtoseconds ffp focal plane position fps frames per second g gravitational constant, or the gravitational acceleration ge degeneracy factors for electrons gi degeneracy factors for ions g0 degeneracy factors for neutral atoms h melt thickness, or specific enthalpy, or Planck’s constant hc convective heat transfer coefficient hch convective heat transfer coefficient inside the hole hpl plasma plume-affected focusing radius hr radiative heat transfer coefficient hrh radiative heat transfer coefficient inside the hole t hi, j,k the value of nodal enthalpy at time t i number of phase steps ) i), )j,k the unit vectors along each of the Cartesian coordinate directions ierfc complementary error function k thermal conductivity, or imaginary part of the refractive index, or 4 extinction coefficient k(T ) thermal conductivity kair thermal conductivity of the air surrounding the workpiece kB Boltzmann’s constant kg the thermal conductivity of the assist gas kl thermal conductivity in liquid state ks thermal conductivity in solid state kv thermal conductivity in vaporization state l thermal diffusion length, or mode number, subscript used in representation of the heat transfer coefficients refer to the sample’s surface with surface normal pointing in the negative (or left) direction of the coordinate axes ld diffusion length ls penetration skin depth lth thermal penetration depth lbwp laser beam waist position m average mass of an evaporation atom, or mass of an electron, or the absolute value of the magnification of the imaging system me mass of melt ejection, or electron mass mm the mass of melted metal that was ejected ms mass of melting of solid metal mv mass of evaporation n real part of the refractive index, or density of a weakly ionized plasma or integer indicating number of orbits in the atom, or the order of the least-squares fit polynomial n1 index of refraction n2 index of refraction ns nanosecond na the electron number density na,g atom number density of gas g nec or nec critical electron density ne electron density or plasma density ng flow velocity of the assist gas ni density of singly ionized atoms n0 total gas density or the neutral ion atom density inside the plasma nZ,g Z-charged ion number density of gas g n complex index of refraction p perimeter of the workpiece, or pressure, or mode number, or position factor or the transform variable p(τ) the normalized beam power ps picoseconds pin pressure component enters the finite element volume pout pressure component exits the finite element volume 5 pb the pressure within the bubble pc the assist gas pressure at the nozzle exit pe the external pressure exerted on the material by evaporating atoms peff the effective static gas pressure pi the assist gas pressure inside the nozzle pr evaporation recoil pressure ps saturated vapor pressure psat(Ts) saturated vapor pressure as a function of temperature p(τ) normalized total beam power as a function of normalized time into the pulse ps picosecond q heat flux or lens shape factor qc convection heat flux qn the magnitude of the heat flux in the n-direction qr radiation heat flux qrad the magnitude of radiation flux r radial coordinate, or direction along the melt surface, or subscript used in representation of the heat transfer coefficients refer to the sample’s surface with surface normal pointing in the positive (or right) direction of the coordinate axes r0 the radial distance at which the surface temperature is equal to the boiling point rb the radius of critical bubble rblur the blur introduced because of diffraction effects rc correlation length rg the assist gas density rm the melt puddle radius rpl half axis width of the plasma plume s the transform variable t time tdisp melt displacement time te the electron cooling time tej the time for melt displacement from the melt puddle tl the lattice heating time tmax the maximum time tp laser pulse duration tpulse the total time of the power meter measurement u dummy variable, or internal energy uo welding speed v linear vaporization rate, or collision frequency vc electron-atom/ion-phonon collision frequency vd drilling velocity vde “drilling” velocity due to ejection of the melt vdm melt surface velocity determined by melt ejection 6 vdv “drilling” velocity due to evaporation of the melt vdvs velocity at which the surface of the condensed phase recedes due to loss of material by vaporization ve melt-ejection velocity vl specific volumes for liquid state vm melt velocity vr radial melt flow velocity averaged over the melt layer thickness vv evaporation velocity, or specific volumes for vapor state vs speed of sound in the solid w diameter of the core of fiber optic cable w beam radius t wspi, j,k a record of the number of vaporized nodes above the node (i,j,k) x,y,z Cartesian coordinates z direction normal to the melt surface, or the spatial Cartesian coordinate z0 axial coordinate of defocusing beam (z)eq the equilibrium distance zf the location of the focal plane zF laser material interaction area zpl half axis length of the plasma plume zthresh depth threshold A surface area, or surface absorptivity, or optic absorbtance Aamb an ambient pressure dependent coefficient Aeff the effective area of flow entering the hole Al the area of the incident laser beam ArF laser material interaction area B the jet plate distance B0 the evaporation constant C arbitrary constant CAZ chemical affected zone CW continuous wave CP critical point CPA chirped pulse amplification D diameter of the lens, or beam diameter on the focusing lens or microscopic displacement or arbitrary constant DOF depth of focus D_ave average diameter of the hole D_entrance entrance spot diameter on the top surface DF diameter of the laser beam in the focal plane of the focusing lens DL laser beam size when it propagates to the front side of the focus objective lens D_m minimum hole diameter Dmin minimum beam diameter that can be achieved 7 Dn is the beam size at location Zn D_exit exit spot diameter on the bottom surface D0 diameter of laser beam on the lens, or diameter of the beam waist DH digital holography E energy, or modulus of elasticity, or electric field, or phasor amplitude of the EM field E0 maximum electric field intensity on axis Eζ,t energy of electrons a considered region Ez,t energy of phonons a considered region EM electromagnetic field EMW electromagnetic wave Ea an activation energy evaporation per atom Edisp the displacement threshold energy Edr drilling threshold energy Eb,λ blackbody monochromatic emissive power Ef an electron at the final continuum state Ei ionization potential for the neutral atoms in the gas at the initial continuum state Em the melting threshold energy Ephonon mean energy of phonon Ereq energy required for micromachining process Es electric field intensity F the focal length of the last focusing optical lens on the path of the laser beam, peak load, or focal number of focusing optic, or factor that depends on the characteristic of the fiber Fth fluence breakdown threshold FDM finite difference model FO Fiber optic G collection of time and space terms of finite difference equation HAZ heat affected zone Hf the latent heat of fusion Hv the latent heat of vaporization Hv0 is the latent heat vaporization at absolute zero I intensity I(z) the intensity of the incident radiation at a given distance, z, into the absorbing medium from the irradiated surface IA numerical approximation of laser beam intensity of the normalized area I0(x,y,0,t) is laser radiation intensity at the material surface Iabs absorbed laser intensity I0(τ) is the