Matlab Laser Toolbox User Manual
Contents
1. datestr Date e g of measurement timestr Time e g of measurement 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 ixy Array of values of I x y It should be noted that some data fields are optional In the case several power density distributions are defined or measured at specified locations z along the optical axis an array of these data structures is constructed Then each entry of the array holds a pdd structure defined in Table 1 one for each specified location z Although arrays of structures in MATLAB are less computationally efficient than structures of arrays the readability and usability of the code of the Laser Toolbox requires the use of arrays of structures Below some functions which create a pdd structure or create it from measured data 2 2 Propagation properties of a laser beam beam Table 2 shows the fields of the data structure referred to here as beam holding propaga tion properties of a laser beam as defined in the 15011146 standard dealing with Lasers and laser related equipment Test methods for laser beam widths divergence angles and beam propagation ratios Table 2 Data structure beam holding propagation properties of a laser beam FIELD DESCRIPTION name Identifier string lambda Laser wavelength m double power Power W of the laser radiation double Zz Locations z along axis of propagation vector of doubles dr Diameters of the beam at location2. 4 Run Matlab s graphical path set tool by executing the command pathtool from the command line and add the directory of the Laser Toolbox here c laser to the MATLAB search path by clicking the Add folder button Or execute the command addpath c laser end from the command line 5 Execute the command help laser from the command line which should give to following output 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 Laser Toolbox Version 0 1 R2007b September 20th 2010 Power density distribution gauss templ temmn tophat rectunif t2surfsrc mdfread uffread plotpdd disppdd Laser beam iso11146 caustic dispbeam Gaussian power density profile Gauss Laguerre mode Gauss Hermite mode Top hat power density profile Rectangular uniform power density profile Surface power density profile based on desired temperature profile Read MDF file from disk Read UFF file from disk Power density distribution plot Display power density distribution Beam propagation ratios according to ISO 11146 1 Caustic plot Display beam characteristics Temperature profile tpntsrc tlinesrc 5 tsurfsrc plottemp General overlap materials momentxy num2pstr hermite laguerre c 2008 2010 G Temperature profile of point surface heat source Temperature profile of line heat source Temperature profile of surface
3. or f t which is fast but inacurrate 3 3 4 PLOTTEMP PLOTTEMP TEMP plots a contour plot and a parametric mesh of the 3D temperature profile at the minimum z corrdinate specified in the struct TEMP as well as corresponding cross sections in the xz plane and the yz plane 17 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 3 4 General functions 3 4 1 OVERLAP OVERLAP D F V plots 10 circular laser pulses of diameter D m at pulse frequency F Hz at velocity V m s of the laser beam relative to the substrate 3 4 2 MATERIALS MATERIALS defines structs with materials parameters and saves these workspace vari able s to disk The structs contain the following fields name Indentifier string A Absorptivity vector of double s lambda Laser wavelength m for which A is defined vector of double s K Thermal conductivity W m K double rho Density kg m 3 double Cp Thermal heat capcity J kg K double kappa Thermal diffusitivity m 2 s double T Temperature K for which the above parameters are defined double Tm Melt temperature K double Tv Vaporization temperature K double 3 4 3 MOMENTXY MX MY VX VY VXY MOMENTSXY X Y Ixy returns the first order moments MX m and MY m in x and y direction of the matrix Ixy W m holding values of power density distribution at the coordinates specified by vectors X and Y as well as the cor
4. If PDD holds three planes or more the following fields are also returned Cx Coefficients of 2nd order polynomial fit of PDD widths vector of 3 doubles zOx Location of focus waist m along optical axis in XZ plane double d0x Width of focus waist m in XZ plane double divx Full fair field divergence angle mrad in XZ plane double zRx Rayleigh length m in XZ plane double M2x Times limited diffraction number beam quality in XZ plane double Cy Coefficients of 2nd order polynomial fit of PDD widths vector of 3 doubles zOy Location of focus waist m along optical axis in YZ plane double 13 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 doy Length of focus waist m in YZ plane double divy Full fair field divergence angle mrad in YZ plane double zRy Rayleigh length m in YZ plane double M2y Times limited diffraction number beam quality in YZ plane double Cr Coefficients of 2nd order polynomial fit of PDD widths vector of 3 doubles zor Location of focus waist m along optical axis double dOr Diameter of focus waist m double divr Full fair field divergence angle mrad double zRr Rayleigh length m double M2r Times limited diffraction number beam quality double 3 2 2 CAUSTIC CAUSTIC BEAM plots the beam dimensions width length diameter of BEAM struct along the optical axis z If three or more planes are defined in BEAM als
5. Thermal Modeling An Introduction to the theory of laser material processing Chapman amp Hall 2001 equation 3 21 15 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 Multi integration The laser intensity profile 7 x y is approximated as piecewise constant with the value x y I x y in the region zi eR e lt a lt aj 4 Bs ae lt eae 14 IN A N ATIN A ESIN representing a square grid of N x N equidistant points in the xy plane L L L L SS E 15 Gwe Sos 7 SU sz 15 The temperature distribution 11 with U 1 can be numerically evaluated as AD A amp E I zi yj v a ri 0 T 2 y 2 t To KN dd g exp 2 16 where Y y 1u xi y yj 27 As V is a factor in the denominator of the function W it represents a singularity which causes numerical problems if Y lt 1 In that case the function W should be replaced by Wo which is defined by 1 27 KAzXAy PE 2 A Rint Cee OE aig Cee 17 w Ax w2 Ay 4z tan tan gt 2en 2z 22 where Ar Ay L N and 42 cos 3 oe 422 sin 8 wy W2 sin 8 cos 3 B t n 22 In the case z 0 and Y lt lt 1 function W should be replaced by lim _ 9 Wo The number of operations which are required to evaluate 16 for a xy plane of N x N nodal points is proportional to N4 This is denoted by O WV For accuracy reasons N must be
6. along the axis of propagation is described by a second order polynomial the fields containing the vectors z dr dx dy and eta should at least hold three elements Only then the other fields of the data structure are defined The function iso11146 of the Laser Toolbox can be used to calculate the data fields of the beam structure from the data fields of the pdd structure 2 3 Temperature profile temp Table 3 shows the fields of the data structure referred to here as temp short for tempera ture holding data related to a 3D temperature profilei e temperature rise T z y z K e g induced by absorbed laser energy In the case a transient temperature distribution is defined at specified time instances t s an array of this data structure is constructed Then each entry of the array holds a temp structure defined in Table 3 one for each every instant t specified Note the similarity of this data structure to the structure pdd of the power density distribution s Table 3 Data structure temp holding 3D temperature profile s FIELD DESCRIPTION name Identifier x x coordinates of T x y z y y coordinates of T x y z Zz z coordinates of T x y z Txy Values T x y z t Time t gt 0 s 2 4 Material Table 4 shows the fields of the data structure referred to here as mat holding material properties It should be noted that some data fields are optional In the case the material properties are temperature dependent an array of t
7. for each plane 3 1 10 DISPPDD DISPPDD PDD displays the characteristics of each plane of a power density profile struct PDD S DISPPDD PDD returns a cell array S of strings of the characteristics of each plane 3 2 Laser beam propagation parameters The 15011146 standard governs test methods for laser beam widths divergence angles and beam propagation ratios 3 2 1 15011146 BEAM 1S011146 PDD returns a strcut BEAM containing the propagation parameters or ratio s according to the international standard ISO11146 1 Lasers and laser related equipment test methods for laser beam widths divergence angles and beam propaga tion ratios part 1 Stigmatic and simple astigmatic beams 2005 with the following optional fields name Identifier string filename Filename string wavelength Laser wavelength m double Zz z coordinates along optical axis vector of doubles mx 1st moment at all planes in x direction vector of doubles my 1st moment at all planes in y direction vector of doubles Vx 2nd moment variance at all planes in x direction vector of doubles vy 2nd moment variance at all planes in y direction vector of doubles dx Width of PDD at all planes in x direction vector of doubles dy Length of PDD at all planes in x direction vector of doubles dr Diameter of PDD at all planes in x direction vector of doubles eta Elipticity dx dy of PDD at all planes in x direction vector of doubles
8. large typically N gt 250 Wo 2 y Z i Yj v 18 Two dimensional Fast Fourier Transform Even fewer operations are required when the two dimensional Fast Fourier Transform FFT algorithm is applied to evalu ate 11 numerically For that purpose it is noted that the steady state version U 1 of equation 11 can be rewritten as 00 00 Tone i 1 Ae gf DWE a adi 19 00 00 16 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 which is a convolution of the intensity profile AJ and the function W with respect to the coordinates x and y i e T AI xW where denotes the convolution operator Hence by applying the two dimensional Fourier transform F2 equation 19 can be rewritten as Fo T FAAI x W AFAI Fo W 20 where the well known equality Fo f x g Fo f Fo g has been applied Finally by applying the inverse Fourier transform to this expression yields T AFy Fo I Fo W 21 By replacing the continuous convolution by discrete convolution and using the two dimensional FFT method the temperature distribution 21 can be calculated numer ically in only O N operations This method was used to calculate the temperature profile in a substrate A basic condition for the two dimensional Fourier transformation is that the function to be transformed is periodic in the ry plane This is not the case for the intensity profile I and the f
9. m PDD TEMPL p 1 P d N returns the struct PDD of which the power density PDD ixy W m is a N by N matrix and the corresponding x and y vectors have length N If not specified N equals 128 PDD TEMPL p 1 P d X Y returns the struct PDD of the power density distribution at locations defined by the vectors X and Y 3 1 3 TEMNN The power density distribution of a Gauss Herimte Transverse Electro Magnetic TEM mode designated by the integer mode numbers m and n in an xy plane is defined by 2242 2m aia exp ee 2 Tmn ay lg ar 2 n 2yv2V2n 1 4y 2n 1 P where H denotes the m order Hermite polynomial dy m and dy m the beam length and width respectively The width and length are defined according to the ISO11146 The intensity scale factor Ip expressed in the modes numbers and the total laser power P reads E 29m 2m 1 2n 1 5 I j nlmlrdsdy 4 The times diffraction limited factor for Gau Hermite modes equals M m n 1 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 PDD TEMmn m n P d returns a structure struct PDD with a laser power density distri bution of a passive laser resonator in Gauss Hermite mode TEMmn with scalar mode numbers m and n power P W and where diameter d m is a 2 element vector with dimensions of the distribution in x and y direction PDD TEMMN m n P d N returns the struct PDD of whic
10. AL SEPTEMBER 23 2010 31 6 T2SURESRE ye a a AA AO ea ae eg oe 11 Silt MDPREAD 25 25 40k gee PoP A A Sk ee Pew Ae 11 Or lis UREREAD G ss a Mo te A ON ee u SES a 12 SAMO PEOT PDD 000 ae aS oe RAE ee ok AAA Ee eo 13 3 1 10 DISPPDD aas i atti A bo de he Gk ee Oe Ee Sed EP da 13 3 2 Laser beam propagation parameters 0000000 e 13 32 1 ISOLDA o a SS a ee AAA a 13 3522 ICAUSTIE Cie sr cog Gy te hk eA ey A E 14 3 297 DISPBEAM pi Epes aces ai Swe ee Bhi eee e 14 3 3 Temperature models e 14 33l IPNTSRE 2 sate ia e a a BIE oe ah ee BEB 14 Ovo Ze TE ENESRG E 8 kt She a ee re i OO Ss a 14 0 J LSURESRG viii toc Slew Si ae A EE Me Bb ae eek ee BO Se 15 3 3 4 PLOTTEMP so o Oe ee a ee a Be dh ee ele 17 3 4 General functions se s 6 gee ee ee ee ka ee eed 18 3 41 OVERLAP 2 2 cose A 2 Bete ee ae A 18 4 27 MATER TALS pce is ck aco o ote had ll Mal che od aon 18 34 3 MOMENTXY n 9 208 oc fo da o LO a ae a ee a Be ek 18 3 44 NUM2PSTR voo o oo WE Ry ae By aes ge oe A A 18 JLo HERMITE 23 56 2 2 ga A te das a BOR Ba ee He BIER i 18 34 69 LAGUERRE 2 83 a2 toe han the Slee of hep hhh Bye Roe ae Slee 19 4 Known issues 19 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 1 Introduction 1 1 Matlab Laser Toolbox MATLAB is high level interpreted language and interactive environment for algorithm development data visualization data
11. Matlab Laser Toolbox User Manual Gert willem R mer UNIVERSITY OF TWENTE Faculty of Engineering Technology Chair of Applied Laser Technology P O box 217 NL7500AE Enschede the Netherlands Email g r b e romerQGwb utwente nl Phone 31 53 4892519 September 23 2010 Version 0 1 beta Preface The Matlab Laser Toolbox provides several functions and scripts for analysis and visu alization of laser beam properties as well as functions to calculate the interaction e g induced temperature by absorbed laser energy in a solid in a material Contents 1 Introduction 3 1 1 Matlab Laser Toolbox 000000000002 eee 3 1 2 System requirements aooaa 3 1 3 Installation instructions ooo e a 3 1 4 Licence and disclaimer 4 2 Data structures 5 2 1 Power density distribution pdd ooo aa ooh ae Ae gal 5 2 2 Propagation properties of a laser beam beam 6 2 3 Temperature profile temp soaa eee eee eee eee 7 2A Maternal eat tne BM heh oon MoU oe Ns Shar Ce Me oe Ae aoa 7 3 Functions by category 8 3 1 Power density distribution 2 20 000 ee ee ees 8 dll GAUSS 02 Son ee A OR Se ee ee 2 SS ef 8 Soleo SPEMBD a ood Aletha sk e a ik ee A oo Phe Ak ne 8 Ss STEMNNS mee Ol Ml eM Sheet GBM e ttn O nO E el S 9 Sele AY IOBHAT ot oa a el e CY eM oe a Na tM Oh hk A 10 dr lo TREGTUNTE soe 2h oa bah eee eA LY ae rng AO ee ee e i 10 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANU
12. analysis and numeric computation Developed by MathWorks Inc http www mathworks com Add on toolboxes which are collections of special purpose MATLAB functions and scripts extend the MATLAB environment to solve particular classes of problems The Laser Toolbox provides several functions and scripts for analysis and visual ization of laser beam properties as well as functions to calculate the interaction e g induced temperature by absorbed laser energy in a solid in a material This User Manual addresses the usage of the Laser Toolbox including examples The author appreciates that publications describing work using the Laser Toolbox quote one of the references given below e G R B E Romer and A J Huis in t Veld Matlab Laser Toolbox Physics Proce dia 5 0 Pages 413 419 e G R B E R mer and A J Huis in t Veld Matlab Laser Toolbox Proceedings of the 29th International Congress on Applications of Lasers amp ElectroOptics ICA LEO September 26 30 2010 Anaheim CA USA Critical evaluation of the Laser Toolbox is welcomed 1 2 System requirements The Laser Toolbox was developed for MATLAB version 7 5 and should run on any operating system supported by MATLAB 1 3 Installation instructions 1 Download the Matlab Laser Toolbox as a compressed ZIP file from the website http www wa ctw utwente nl software laser 2 Decompress the ZIP file into a directory on your hard disk e g in c laser 3 Start MATLAB
13. ative to the surface exp Ko eee 10 T x y 2r K 2K 2K where K denotes the thermal conductivity W m K and the thermal diffusitivity m s of the material And Ko is the modified Bessel function of order 0 T TLINESRC MAT Q V X Y returns the 2D temperature profile T at x and y coordinates defined by vectors X and Y in a semi inifite material MAT due to a line heat source of Q W m moving at velocity V m s in the positive x direction relative to the material 3 3 3 TSURFSRC Assume a semi infinite substrate with constant material parameters and a surface heat source absorbed laser energy defined by the power density I x y W m moving over the substrate s surface This surface heat source will induce the following 3D temperature rise in the substrate when it is moving at a constant velocity of v m s in the x direction relative to the surface of renee f CA AOR CRT RUE 11 00 00 where R y 12 y 22 and exp a e 4 R 12 1 W Y Z E ys V 2TKR and U R t v 1 exp LEA de 1 V at O in which K denotes the thermal conductivity W m K and the thermal diffusitivity m s of the material This expression can be evaluated numerically in two ways multi integration and using the Fast Fourier Transform FFT For simplicity the steady state situation t oo so U 1 is considered only below 13 J M Dowden The Mathematics of
14. h the power density PDD ixy W m is a N by N matrix and the corresponding x and y vectors have length N If not specified N equals 128 PDD TEMMN m n P d X Y returns the struct PDD of the power density distribution at locations defined by the vectors X and Y 3 1 4 TOPHAT A Top Hat power density distribution in an xy plane is defined by Hey Pre Vere lt L Y 0 y 12 y gt where d m is the diameter of the power density profile and P W the total laser power 5 NIQ NIA PDD TOPHAT P d returns a structure struct PDD with a Top Hot power density distri bution of power P W and diameter d m PDD TOPHAT P d N returns the struct PDD of which the Top Hat power density PDD ixy W m is a N by N matrix and the corresponding x and y vectors have length N If not specified N equals 128 PDD TOPHAT P d X Y returns the struct PDD of the power density distribution at loca tions defined by the vectors X and Y 3 1 5 RECTUNIF A Rectangular Uniform power density distribution in an xy plane is defined by I rane x y ER z lt dz amp y lt dy x y y 0 elsewhere 6 where dy m and dy m are the dimensions of the power density profile and P W the total laser power PDD RECTUNIF P d returns a structure struct PDD with a rectangular uniform power density distribution of power P W and where diameter d m is a 2 element vector with dimensions of the distribution in x and y directio
15. heat source Temperature plot Overlap plot of pulses Sets and saves materials parameters First and second order moments Convert numbers to a prefixed string Hermite polynomial Laguerre polynomial R B E Romer University of Twente This Matlab Laser Toolbox is for private non commercial single home computer or educational use only The use of Matlab Laser Toolbox for commercial purposes is strictly prohibited Please read the detailed license agreements in the Matlab Laser Toolbox User Manual 1 4 Licence and disclaimer Licence The Matlab Laser Toolbox is free for private non commercial single home computer or educational use only The use of Matlab Laser Toolbox for commercial purposes is strictly prohibited By downloading the Laser Toolbox you express your approval of the conditions and terms hereof and you are bound thereby Should you disagree with the 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 conditions and terms hereof promptly terminate the use of this software and destroy any copies thereof or delete the program already downloaded The Laser Toolbox subject thereto is and continues to be a property of the University of Twente The Laser Toolbox is also subject to copyrights and as such it is subject to a full protection thereof Disclaimer The Matlab Laser Toolbox is provided as is without warranty of any kind express or implied including but not li
16. his data structure is constructed Then each entry of the array holds a material structure defined in Table 4 one for each temperature Note that regarding this aspect the similarity of this data structure to the data structures pdd and temp Table 4 Data structure material holding material properties FIELD DESCRIPTION A Absorption 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 lambda Laser wavelength s scalar or vector of doubles n Refractive index indices k Extinction coefficient s rho Density kg m3 K Heat conductivity W mK Cp Heat capacity J K Tm Melt temperature K Tv Vaporisation temperature K defined at wavelengths specified by lambda 3 Functions by category The functions of the Laser Toolbox are easy to analyze and easily modified in the case the user needs to extend or adapt the functionality of the scripts In the remainder of this section the usage of these functions are illustrated by examples 3 1 Power density distribution 3 1 1 GAUSS A Gaussian power density distribution in an zy plane is defined by 2 Ies p 0 22 2249 1 md where P W is the laser power and d m the diameter of the power density distribution PDD GAUSS P d returns a structure struct PDD with a Gaussian power density distri bution of power P W and diameter d m PDD GAUSS P d N returns the struct PDD of which the Gaussian power density PDD ixy W
17. ixy px py matrix power density W m 2 If FILENAME is ommitted a dialog box for the user to select a filename is displayed 3 1 8 UFFREAD PDD UFFREAD FILENAME reads the data from a file D created by the Laserscope by PROMETEC GMBH representing the power density profile caustic of a laser beam measured by the Laserscope UFF100 of PROMETEC GMBH http www prometec de and returns a PDD stuct with the following fields name Prometec UFF100 Laserscope string filename File ID string comment Comments string px No of pixels along x axis integer Py No of pixels along y axis integer focus Integer amp Amplification of the sensor signal integer windowsize Window size integer windowcenterx x coordinate of window center 48x1 double windowcentery y coordinate of window center 48x1 double versionsensor Version sensor integer versionuff Version UFF integer power Laser power W datestr Date string timestr Time string x x coordinates vector of px elements y y coordinates vector of py elements ixy px py matrix power density W m 2 If FILENAME is ommitted a dialog box for the user to select a filename is displayed 12 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 3 1 9 PLOTPDD PLOTPDD PDD plots a contour plot two cross section as well as a parametric mesh of the power density profile s in the struct PDD one figure
18. m is a N by N matrix and the corresponding x and y vectors have length N If not specified N equals 128 PDD GAUSS P d X Y returns the struct PDD of a Gaussian power density distribution at locations defined by the vectors X and Y 3 1 2 TEMPL The power density distribution of a Gauss Laguerre Transverse Electro Magnetic TEM mode designated by the integer mode numbers p and l in an xy plane is defined by 8r M 8r M2 8r M mo to Lp ER e 2 2 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 where Jo W m denotes the intensity scale factor d the diameter of the profile and M gt 1 the times diffraction limited factor or short beam quality number The beam diameter and the beam quality number are defined according to the 15011146 standard Finally L denotes the generalized Laguerre polynomial of order p and index l The intensity scale factor To expressed in the modes numbers and the total laser power P reads 2 SMP 1 0 Td lo 2 3 SOME PAG Be mrd p 1 y For Gau8 Laguerre modes the times diffraction limited factor equals M 2p 1 1 In stead of polar coordinates r y carthesian coordinates are returned by the function TEMPL PDD TEMPL p 1 P d returns a structure struct PDD with a laser power density distri bution of a passive laser resonator in Gauss Laguerre mode TEMpl with scalar mode numbers p and 1 power P W and diameter d
19. mited to the warranties of merchantability fitness for a particular purpose and noninfringement In no event shall the University of Twente nor the authors or copyright holders be liable for any claim damages or other liability whether in an action of contract tort or otherwise arising from out of or in connection with the software or the use or other dealings in the software 2 Data structures Four data structures hold and organize related data to be handled and processed by the Laser Toolbox 2 1 Power density distribution pdd Table 1 shows the fields of the data structure referred to here as pdd an abbreviation of power density distribution This structure holds data related to a power density distribution I x y W m measured or calculated at a specified plane along the axis z of propagation of the laser beam Table 1 Data structure pdd holding power density distribution s FIELD DESCRIPTION name Identifier fileid ID of measurement file if any comment Comments if any px No of values in x direction of I x y py No of values in y direction of I x y rangexy Length m and width m of xy plane x x coordinates at which I x y is defined y y coordinates at which I x y is defined Zz z coordinate at which I x y is defined amp Amplification dB of sensor if any average No of measurements of I x y offset Offset noise level of sensor lambda Laser wavelength m foclen Focal length of lens used if any
20. n PDD RECTUNIF P d N returns the struct PDD of which the power density PDD ixy W m is a N by N matrix and the corresponding x and y vectors have length N If not specified N equals 128 10 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 PDD RECTUNIF P d X Y returns the struct PDD of the power density distribution at locations defined by the vectors X and Y 3 1 6 T2SURFSRC Besides the calculation of the temperature distribution given the intensity profile J and the function W see equation 11 also the inverse problem can be solved by applying the FFT method see section 3 3 3 That is given a desired temperature distribution T and the function W the required intensity profile J inducing the desired temperature field can be determined by rewriting equation 20 as 1 PAD FATHER WIN 7 and by applying the inverse Fourier transform Then the intensity profile reads Leys z I FF PAT FAW a 8 This is a very powerful method as it allows the calculation of the required intensity pro file directly from any desired temperature distribution It is evident that this method will only yield a feasible intensity profile I x y gt 0 Y x y R if the desired tem perature field is physically feasible PDD T2SURFSRC MAT T V returns a PDD struct containing a power density distribution which when absorbed at the surface z 0 generates the 2D temperature profile struc
21. o the corre sponding 2n order polynomial fits are plotted 3 2 3 DISPBEAM DISPBEAM BEAM displays the characteristics of laser beam struct BEAM S DISPPDD BEAM returns a cell array of strings of the characteristics of the laser beam 3 3 Temperature models 3 3 1 TPNTSRC Assume a semi infinite substrate with constant material parameters and a point source of heat of P W on the substrate s surface This line heat source will induce the following 2D temperature rise in the substrate when it is moving at a constant velocity of v m s in the x direction relative to the surface Te 24 exp H Vyr y 2 9 27K fx y2 2 where K denotes the thermal conductivity W m K and the thermal diffusitivity m s of the material T TPNTSRC MAT P V X Y returns the 3D temperature profile T at x y and z coordinates defined by vectors X Y and Z in a semi inifite material MAT due to a point heat source of P W moving at velocity V m s in the positive x direction relative to the material 3 3 2 TLINESRC Assume a semi infinite substrate with constant material parameters and a semi infinite line source of heat of Q W m perpendicular to the substrate s surface This line heat source will induce the following 2D temperature rise in the substrate when it is moving 14 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 at a constant velocity of v m s in the x direction rel
22. responding second order moments variance VX m and VY m and cross variance VXY This function is used by 18011146 3 4 4 NUM2PSTR T NUM2PSTR x converts the scalar x into a string representation T with about 4 digits and a prefix of the metric system This is useful for labeling plots with the TITLE XLABEL YLABEL and TEXT commands 3 4 5 HERMITE The ntt degree Hermite polynomial H reads HOE lt 2x n 22 18 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 Y HERMITE N X returns the Nth degree Hermite polynomial of X where N is a non negative integer and X a real scalar This function is used by TEMMN 3 4 6 LAGUERRE The Laguerre polynomial L of order n gt 0 and index l gt 0 reads 1 n 0 Ley 1 r l m 1 1 Qn l 1 2 LP a Un 1 DL 2 n gt 1 23 Y LAGUERRE N L X returns the generalized Laguerre polynomial of order N and index L of X where N and L are a non negative integers and is X a real scalar This function is used by TEMPL 4 Known issues 1 Some scripts can will be optimized for speed even more 19
23. s z vector of doubles dx Width of the beam at locations z vector of doubles dy Length of the beam at locations z vector of doubles eta Ellipticity of the beam at locations z vector of doubles dOr Diameter of waist focus of beam double zor Location at optical axis of waist focus double zRr Rayleigh length of beam double divr Far field divergence angle rad of the beam double M2 Beam propagation ratio quality M2 double Cr Coefficients of 2nd order polynomial describing beam diameter propagation vector of 3 doubles dOx Diameter of waist focus in xz plane double zOx Location of waist focus in xz plane double ZRx Rayleigh length of beam in xz plane double divx Far field divergence angle rad in xz plane double M2x Beam propagation ratio M2 in xz plane double Cx Coeff of 2nd order polynomial describing diameter propagation in xz plane vector of 3 doubles doy Diameter of waist focus in yz plane double 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 zOy Location of waist focus in yz plane double zRy Rayleigh length of beam in yz plane double divy Far field divergence angle rad in xz plane double M2y Beam propagation ratio M2 in yz plane double Cy Coeff of 2nd order polynomial describing diameter propagation in yz plane vector of 3 doubles This datastructure allows also the characterization of simple astigmatic laser beams As the propagation of the beam diameter
24. t T in a semi inifite material MAT when the PDD moves at velocity V m s relative to the material The x and y dimensions of PDD are defined by the x and y coordinates of PDD 3 1 7 MDFREAD PDD MDFREAD FILENAME reads the data from a file extension mdf created by the LaserDiagnoseSoftware version 2 81 by PRIMES GMBH representing the power den sity profile s caustic of a laser beam measured by the FOCUSMONITOR of PRIMES GMBH http www primes de and returns an array of PDD stucts one for each plane with the following fields name FocusMonitor string filename Filename string fileid File ID string comment Comments string px No of pixels along x axis integer Py No of pixels along y axis integer rangexy Row vector with measuring range in x m and y m direction Zz z position m along the axis of propagation posxy Row vector with x m and y m position of measuring range amp Amplification of the sensor signal dB double 11 2010 G R B E ROMER MATLAB LASER TOOLBOX USER MANUAL SEPTEMBER 23 2010 average Number of averages integer offset Offset noise level of data integer lambda Wavelegth laser m power Laser power W foclen Focal length m datestr Date string timestr Time string dataxy Data px py matrix integers x x coordinates vector of px elements y y coordinates vector of py elements dataxy Data px py matrix integers
25. unction W and will introduce errors in the evaluation of the temperature distribution These errors are introduced by the truncation of J and W i e if L grid size is chosen too small then for z gt L and y gt L W x y 4 0 and I x y 0 The errors are significant for low temperatures only To reduce these errors the region of calculation should be chosen sufficiently large As a rule of thumb one should choose L gt 3d where d is the diameter of the laser beam T TSURFSRC MAT PDD V returns steady state time is infinity 3D temperature profile T of a struct TEMP in a semi inifite material MAT due to a surface heat source defined by a single plane in PDD moving at velocity V m s relative to the material The x and y dimensions of T are defined by the x and y coordinates of PDD The z coordinates depth are 0 d 2 and dp m by default where dp is the heat penetration depth T TSURFSRC MAT PDD V TIME returns an array of temperature structs T one struct per time instance defined by vector TIME If not defined TIME Inf T TSURFSRC MAT PDD V TIME Z returns an array of temperature structs T one struct per time instance defined by vector TIME at z coordinates m defined by vector Z T TSURFSRC MAT PDD V TIME Z METHOD returns an array of temperature structs T at time instances at z coordinates m Z using the calculation method METHOD METHOD is a string equalling mulitint default which slow er but accurate
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