IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 Mathematical Model of Effect of Number of Pulses of Pulsed Laser o n Formation Process of Plasma S. K.Mousa Departme nt of Physics, College of Education for pure science, Unive rsity of Anbar Received in Oct. 5 2010 Accepte d in Feb. 8 2011 Abstract The effect of number of p ulses of p ulsed laser on materials is st udied analytically, different p ulses has been used with the same delay time. The depth of p ossible damage to the surface of copp er and titanium as well as depth of the crater to both materials were considered in this study . The study revealed that linear model is only p ossible when estimating depth of p ossible damage for copp er material, this means that the depth of p ossible damage increases with the increment of number of laser p ulses .As for titanium material, it is found the relationship is nonlinear. The depth of p ossible damage of titanium and copp er is not the same, and copp er seems t o be more predictable than titanium. k ey words: Pulsed laser, delay time, linear models, copp er. Introduction Since the discovery of the laser, researchers began irradiating almost every conceivable target material and p hase for novel basic p hy sical and chemical st udy in addition to new app lications. Pulsed-laser deposition (PLD) technique has been successfully app lied to a wide range of materials. In the laser ablation p rocess, high-p ower laser p ulses are focused onto the surface of a bulk target ty p ically at an angle of incidence of 45° with resp ect to its p erpendicular direction. The ablation of the target p roduces a visible p lasma (commonly known as "p lume") that exp ands from the target[1]. The production of high temp erature p lasmas with small scale, short p ulse, high intensity lasers has been actively p ursued during the last three decades. Of p articular interest in these st udies is the measurement of the energy absorp tion efficiency of high density p lasmas created by intense irradiation of a solid target, and many group s have p ublished work measuring solid target p lasma absorp tion efficiency over a wide range of incident intensities and laser wavelengths on p lanar[2-6] and micro st ructured targets[7]. These st udies have shown that t he plasma ty p ically absorbs a large fraction of the laser energy , between 10% and 80% of the incident energy , depending up on intensity and laser wavelength. Such exp eriments have shown that a large amount of energy can be deposited p er unit area and that high temp eratures about 100 eV are achievable[2,8]. However, rapid heat conduction into t he cold, solid subst rate beneath the p lasma will ty p ically clamp the p lasma temperature to a value of 1000 eV[8-10]. Furt hermore, most of the deposited energy is contained within the p lasma electrons, which cool too rapidly by conduction and hy drody namic exp ansion to transfer much of this laser energy to the cold ions. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 Data collection A rough idea of the size of a crater can be obtained by considering the following: Pressure is the energy p er unit volume, P = dE/dV ≈ E/V, where E is the energy of the p ulse delivered to the material and V is the volume of the p lasma (or the gas that condenses from the plasma). If the p ressure is above the st rength of the material K, t he material give way as it is p ushed aside by the p lasma. We can therefore find the app roximate volume of the excavated crater by setting E/V = K. Assuming the crater is a cy linder with a dep th equals to its radius, t he radius of the crater will be ap p roximately Rc = (E/(π K))1/3. The software that is designed for estimating beam p arameters of p ulse energy (M aterials resp onse to p eak intensity .htm) is used here for simulation p urp oses of this p aper. This p rogram is an interactive use interface which can be used directly from the internet (how to build a laser death ray.htm).The user interface of p rogram contained two main categories ;the first is that deals with the ty p e of material and some of the laser p arameters of interest, the second category contained some imp ortant p arameters of material. Both categories lead to the same outp ut. The p rogram has been run frequently in this research according to the need due to varying number of p ulses as well as delay time. Results For both copp er and titanium material, p ulsed laser (x) will be considered as an indep endent variable while the depth of p ossible damage and hole depth were considered as dependent variables denoted by y 1 and y2 resp ectively. The copp er material will be considered first and the simple linear regression equation of y 1 on x was p erformed. The slop e and the intercep t p arameters of the line were found to be significantly different from zero. The coefficient of determination for the equation was 100%, which means absolute straight line[11]. xy 006.0226.01  … (1) Figure (1)shows the scatter p lot of the data and the imp el linear regression line that is the best fit for t he data of these two variables. It is therefore, possible to p redict t he dep th of p ossible damage in cop p er material if the number of p ulses is given in advance. The linear relationship between number of p ulses and hole depth of copp er material made is given in equation (2). This equation clearly shows that relationship between hole depth and number of p ulses is not linear since the coefficient of determination was found to be 0.08 and both p arameters of equation (2) are not significantly different from zero. xy 453.0321.39 2  … (2) Figure(2) revealed that sp line interp olant is the best fit for this set of data. The red line that represents equation (2) gives a fair idea about how bad is the linear representation ,i.e. equation(2)cannot be used for p rediction p urp oses. It is of interest to invest igate whether hole dep th can and dep th of p ossible damage for copp er material can be p ut in a form of linear relationship . Equation (3) shows this relation. In this equation both p arameters are found to be not significantly different from zero and the coefficient of determination equals to 0.08. It is therefore can be concluded that t he relation is far from linear. Figure (3) shows the scatter p lot of the points as well as the fitted models[11]. 12 394.74506.22 yy  … (3) IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 The above p rocedure was repeated in the same manner for the titanium material. In this context, equation (4) shows the simple linear regression line of depth of p ossible damage on p ulse number. xy 606.3972.54 1  … (4) Although the coefficient of determination equals 0.68, but the regression is significant and both p arameters of the equation re significantly different from zero. Figure (4) shows the scatt er p lot of the p oints of these two variables as well as the fitt ed models. It is clear the sp line interpolant gives t he best fitt ing. The simple linear regression equation (5) of the depth hole on the number of p ulses shows that the slop e p arameter is not significantly different zero. And that 53% of the regression line can be interpreted by the intercept[11]. xy 201.3209.64 2  … (5) Figure (5) shows the scatter plot of the points as well as the fitted models. T he sp line interp olnt is t he best fitt ing for this set of data. The simp le linear regression line (equation 6) was found to the worst model fitt ed the data of linear regression y 2 on y 1 . Both p arameters of the equation were to be not significantly different from zero. 12 018.0721.28 yy  … (6) Figure(6) shows that neither the simple linear regression nor the quadratic model can fit t his data. M any att emp ts were made to find the best fit for t his set of data. The quadratic model, although it is worse but it is much better than the linear fitting. Discussion The variability of number of p ulses p rop agated significantly affected the depth of p ossible damage and hole depth of materials (cop p er and titanium) considered in this st udy . It is wort hwhile st ating that t he effect of p ulses on the cop p er material behaved differently when comp ared to titanium. In the first case the damage increased st eadily , whereas in the case of titanium st arted with high damage and tends to decreases with high p ulse numbers. For this reason t he linear relationship in the first was absolutely st raight, whereas it wasn't the state for the second case. The slop e of the hole depth for both materials was not significantly different from zero, that is linear model is not suitable. The titanium material tend to have an app roximately a fixed effect since the intercep t p arameter was found to be significantly different from zero, whereas it wasn't the state with regard to cop p er material. In both materials, it was not p ossible to find a model that govern the relationship between the dep th of p ossible damage and hole depth. Conclusions It is obvious from the all that the copp er material is bett er than titanium in accordance with exp loration the range of damage generated by p ulsed laser in the field of a certain number of p ulses .As for titanium, the behavior was different, this is may be due to the IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 characterist ics of the used laser which not fit with the characterist ics of titanium material. Thus we can take in consideration and examine titanium material to a different ty p es of lasers in order to know t he behavior of this material with the other kinds of lasers. The study illustrated that t he increment of number of pulses during certain p eriods leads t o an essential damage on the materials with no care to the nature of resp onse of the materials(linear or nonlinear) with the number of pulses. Re ferences 1. Smausz ,T.; Hop p , B.; Kecskeméti, G. and Bor Z.(2006). Study on metal micro p article content of the material transferred with Absorbing Film Assist ed Laser Induced Forward Transfer when using silver absorbing lay er” Ap p lied Surface Science, 252, Issue 13, 30 : 4738-4742. 2.M ilchberg,H.M .;Freeman,R.R.;Davey ,S.C.and M ore,R.M .(1988)."Resistivity of a Simp le M etal from Room Temperature to 10 6 K", Phy s. Rev. Lett. 61: 2364. 3.Fedosejevs,R.;Ot tmann,R.;Sigel,R.;Kühnle,G.;Szatmari,S. andSchäfer,F.P.(1990). Absorp tion of femtosecond laser p ulses in high-density p lasma", Phy s. Rev. Lett. 64 : 1250. 4.Ditmire,T.;Gumbrell,E.T.;Smith,R.A.;M ountfrd,L. andHut chinson,M .H.R.(1996). Sup ersonic Ionization Wave by Radiation Transp ort in a short –p ulse- p roduced p lasma,Phy s.Rev.Lett.77:498. 5.Price,D.F.;M ore,R.M .;Walling,R.S.;Guethlein,G.;Shepherd,R.L.;Stewart,R.E.and White,W.E. (1995). Absorp tion 0f Ultrashort Laser Pulses by Solid Targets Heated Rabidly to Temp eratures 1-1000ev, Phy s. Rev. Lett. 75: 252. 6.M egan Jaunic;Shreyaraje;Ky unghan Kim and Kunak M itra (2008). Bio-heat transfer analysis during short p ulse laser irradiation of tissues", Int ernational and journal of heat and mass transfer. 51: 551. 7.Gop alenduap al;Soumyadip ta Basu;Kunal M itra and Tuan Vo-Dinh.(2006). Time –resolved op tical tomography using short -p ulse laser for tumor detection, Ap p lied Phy sics. 45:6270. 8.Az iz,M ichael.(2008). Film growt h mechanisms in pulsed laser dep osition, J. Ap p lied Phy sics A 93: 579. 9.Jep p e By skov-Nielsen (2010). short -p ulse laser ablation of metals: fundamentals and app lications for micro-mechanical interlocking, PhD thesis, Dep artment of Phy sics and Astronomy, University of Aarhus, Denmark. 10.M ozgovoy ,A.G. (2002). A new method of obt aining high temp erature plasma , IEEE Xp lore 1:537. 11.Norman R.and Drap er;Harry Smith(1998). Ap p lied Regression Analysis",3 rd edition, John Wiley &sons.,Inc.,NewYork. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 Fig.(1): S catter plot and fitted equati on for number of pul se s and depth of possible damage (Copper) Fig.(2): S catter plot and fitted equati on for number of pul ses and hole depth (Copper) 2 4 6 8 10 12 14 16 18 20 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 Real data Equation 1 Number of pulses D ep th o f p o ss ib le d am ag e (m m ) 2 4 6 8 10 12 14 16 18 20 -40 -20 0 20 40 60 80 10 0 12 0 Real data Spline Interpolant Equation 2 Number of pulses H o le d ep th ( µ m ) IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 Fig.(3): S catter plot and fitted equati on for depth of possible damage and hol e depth(Copper). Fig.(4): S catter plot and fitted equati on for number of pul se s and depth of possible damage (Titani um) 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0 10 20 30 40 50 60 70 80 90 10 0 Real data Spline Interpolant Equation 3 Depth of p ossible damage (mm) H o le d ep th ( µ m ) 2 4 6 8 10 12 14 16 18 20 -20 0 20 40 60 80 10 0 12 0 Real data Spline Interpolant Equation 4 D ep th o f p o ss ib le d am ag e (m m ) Number of pulses IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 Fig.(5): S catter plot and fitted equati on for number of pul ses and hole depth (Titanium) Fig.(6): S catter plot and fitted equati on for depth of possi ble damage and hole depth(Ti tanium) 2 4 6 8 10 12 14 16 18 20 -20 0 20 40 60 80 10 0 12 0 Real data Spline Interpolant Equation 5 Number of pulses H o le d ep th ( µ m ) 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 10 0 Real data Quadrati c Equation 6 Depth of p ossible damage (mm) H o le d ep th ( µ m ) 2011) 1( 24المجلد مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة عملیات تشكیل فينموذج الریاضي لتأثیر عدد نبضات اللیزر النبضي اال البالزما سالم خلف موسى جامعة االنباركلیة التربیة للعلوم الصرفة،،،قسم الفیزیاء 2010تشرین االول 5 في استلم البحث 2011شباط 8 في قبل البحث الخالصة زمن ب ت نبضات مختلفةاستخدمفقد ،في الموادنبضات لیزر نبضي تأثیر عددل تحلیلیة دراسة أجریت في هذا البحث عمق إلى عن" فضالعمق التلف الممكن على سطح النحاس والتیتانیوم االعتباربنظر في هذه الدراسة أخذ.نفسه التأخیر نموذج الخطي یكون ممكنا فقط في حالة تقدیر عمق التلف الممكن حصوله لقد أظهرت الدراسة أن اال.المادتین الحفرة لكال ن العالقة االدراسة أظهرتبینما .وهذا یعني ان عمق التلف الممكن یزداد مع زیادة عدد نبضات اللیزر،لمادة النحاس وأن مادة النحاس ،مادتي التیتانیوم والنحاس یختلفل ان عمق التلف الممكن.مادة التیتانیومالى تكون غیر خطیة بالنسبة .تبدو أكثر قابلیة للتنبؤ بسلوكها من مادة التیتانیوم .النحاس،النماذج الخطیة،زمن التأخیر،اللیزر النبضي :كلمات مفتاحیة