Beam Characterization of a Materials Processing CO 2 Laser

Transcription

1 WELDNG RESERCH SUPPLEMENT 47-s Beam Characterzaton of a Materals Processng CO 2 Laser Measurements show that tradtonal pano-convex lenses used n laser materals processng do not offer optmum performance M. ESSEN ND P. W. FUERSCHBCH BSTRCT. The propagaton of focused laser radaton has been characterzed for 12 lenses usng the laser propagaton equaton, yeldng measurements of beam qualty, spot sze, and depth of focus. The beam qualty of the laser places an ntrnsc lmt on the mnmum focused spot sze and was found to be M 2 = 1.5 _+0.09 (M 2 = 1.0 corresponds to an deal beam) for the ] 200-W CW CO 2 laser used n ths study. s the lens focal length was decreased, measured values of spot sze and M 2 were sgnfcantly ncreased over the values predcted for aberraton-free lenses. Ths ncrease s beleved to be due to sphercal aberraton ntroduced by the lenses. t was also found that at short focal lengths, aspherc lenses produce better results than panoconvex, menscus, or dffractve lenses, at a laser power of 300 W. Equatons were determned that accurately predct beam qualty, spot sze, and depth of focus under the nfluence of sphercal aberraton. ntroducton Characterzaton of the propagaton and focusng ablty of a laser beam s essental n laser materals processng applcatons. Ths characterzaton s of partcular mportance n processes such as laser drllng and meltng of metals, whch exhbt a well-defned ntensty threshold, below whch the desred result s not obtaned (Ref. 1 ). ndeed, n the nteracton of metals wth powerful lght M. ESSEN and P. W. FUERSCHBCH are wth Sanda Natonal Laboratores, lbuquerque, N. Mex. fluxes, t has been shown that the absorpton of laser energy by metals s ntensty dependent (Refs. 2, 3). s a result, processes such as laser cuttng and drllng, whch requre removal of materal through evaporaton, and solderng and weldng, whch requre meltng, depend strongly on the characterstcs of the materals processng laser. n many cases, the requred ntensty threshold for a materal may be reached by focusng the laser radaton. Consequently, n materals processng applcatons, one should ask: 1 ) what parameter determnes the ntrnsc lmt of the value to whch a laser can be focused; 2) what lmts the focusng ablty of a hgh-qualty laser lens; and 3) how does ths lmtaton vary among dfferent shapes of lenses? requste to an understandng of laser materals processng applcatons s therefore an understandng of the propagaton of laser energy through optcal systems, as well as knowledge of how the beam s affected by the system. Four mportant pa- KEY WORDS CO 2 Laser spherc Beam Qualty Depth of Focus Dffractve Menscus Sphercal berraton Spot Sze Pano-Convex Lens rameters that characterze a laser for materals processng are mnmum spot sze, beam qualty, the mnmum spot sze locaton, and depth of focus. Spot Sze and Beam Qualty n order to defne spot sze (the sze of the laser beam at an arbtrary locaton along ts axs of propagaton), we frst note that the beam profle of a laser operatng n ts fundamental mode has a Gaussan ntensty dstrbuton of the form (Ref. 4): (x,y,z )= loexp[_2(x2w2( z y2 ) (1) where 0 s the maxmum ntensty and w(z) s the spot sze, or the radus of the beam at the pont where the ntensty has dropped to e 2 (13.5%) of the maxmum ntensty. Therefore, a cross-secton of a Gaussan beam taken n a plane P perpendcular to the drecton of propagaton s a crcle descrbed by x 2 + y2 = w2(z), where x and y are orthogonal coordnates n P. When the beam s focused by a lens wth focal length, f, ts dvergence angle ncreases whle the spot sze decreases not to an nfntesmal pont, but to a mnmum value near the focal plane (the plane perpendcular to the optc axs at the focal pont of the lens). The product of the dvergence angle and the mnmunl spot sze has a mnmum value gven by (Ref. 5): M2,L woo - (2) where w 0 s the mnmum spot sze, 0 s the half-angle dvergence of the beam,

2 48-s FEBRURY 1996 REL BEM 2"0 w(z)=w z~r~w~ (8a) (z_ z,) 2 The equvalent expresson n M 2 s therefo re z 0 DEL BEM Fg Propagaton of real and deal focused laser beams. The prmes refer to the real beam. M 2 s the beam qualty, and ~, s the wavelength of the laser radaton. The mnmum possble value of ths product (.e., Wo0 ) occurs when M 2 = 1.0. laser operatng n the fundamental mode (TEMoo) wll have the smallest al- lowed value of WoO = kx (.e., M 2 = 1.0). Ths lmtng value s a result of dffracton of the laser beam, and s analogous to the dffracton of a bundle of rays passng through a crcular aperture wth a dameter equal to the dameter of the unfocused beam. Such a system s sad to be dffracton lmted. Ths however s an deal case, and the beam qualty of a real beam wll always be greater than unty. Physcally, the mnmum product rule causes a real beam to be focused to a spot sze whch s larger than an deal beam focused by the same lens -- Fg. 1. Note that the cross-secton of the beam along the propagaton axs s a hyperbola and may be descrbed by the equaton w 2 _1 a 2 b 2 (3) where a and b are constants and z 0 s the locaton of the mnmum spot sze. The smallest value of w(z) corresponds to the mnmum wast radus. The asymptotes are nclned to the propagaton axs at an angle gven by ~w0 (4) n expresson for the varaton of spot sze along the axs of propagaton may therefore be determned by solvng for the constants a and b usng the general condtons of a laser beam wth an unfocused radus of W 0 focused by a lens of focal length f. more mathematcally rgorous dervaton s gven by Kogelnk and L (Ref. 6). The frst condton s that the laser s focused a dstance z 0 from the lens. Therefore, at z = z 0, w = w 0. (One should note that the value of z 0 s not necessarly the focal length of the lens, snce the laser has a fnte dvergence before enterng the lens.) Substtutng these values nto Equaton 3 yelds a-' : w~ (5a) The second condton s that w(z) = W 0 at the lens. Ths gves _ WoZo (sb) Substtutng a and b nto Equaton 3 and usng W 0 ~> w 0 t wozo (6) Solvng Equaton 2 for M 2 and substtutng 0 = Woz o, the beam qualty may be wrtten as M 2 - ztwowo Zz (7) Rewrtng Equaton 6 to nclude the fac- tor of (M2) 2 yelds FOCUSNG LENS ROTTNGNEEDLE (8b) Equaton 8b s the laser propagaton equaton and s dentcal to the expresson gven by Kogelnk and L (Ref. 6). Therefore, one can determne M 2 usng Equaton 7 by measurng W 0, locatng z o, the poston of the beam wast, and makng a sngle measurement of w 0. lternatvely, values for w 0, M 2, and z 0 may be determned smultaneously by makng measurements of beam radus along the propagaton drecton and fttng to the data a curve of the form of Equaton 8b, wth w 0, M 2, and z 0 taken as the parameters of a nonlnear ft. Spot Sze and Sphercal berraton The focusablty of a laser beam s not only lmted by the beam qualty of the laser, but also by the optcal element(s) used to transmt the beam. Specfcally, the mnmum spot sze of a laser beam focused usng a hgh-qualty laser lens wll be lmted by thrd-order sphercal aberraton. Sphercal aberraton s defned as a varaton of focus wth aperture (Ref. 7), and s nherent n all smple lenses. The two lens parameters that contrbute most to sphercal aberraton n smple lenses are the lens focal length and shape factor. s the lens focal length decreases, the curvature of the lens surface must necessarly ncrease. Ths ncreased curvature causes a ray n the outer porton of the laser beam (a margnal ray) to undergo DETECTOr ~"~'~.-...,~ Fg Schematc of the Laserscope scannng aperture system. FEE D SLDE j

3 WELDNG RESERCH SUPPLEMENT 49-s margnal ray) to undergo more refracton than a ray near the optc axs of the lens (paraxal ray). Consequently, margnal rays wll be focused closer to the lens than paraxal rays. The result of ths phenomenon s to ncrease the mnmum spot sze. The shape factor s also crucal n smple lenses snce the amount of sphercal aberraton produced by a lens may be decreased by changng the shape of the lens. The four shapes or types of smple lenses tested n ths work are pano-convex, menscus, aspherc, and dffractve lenses. Pano-convex and menscus lenses are sphercal lenses,.e., both surfaces of the lenses are segments of spheres and can be defned by a constant radus of curvature. spherc lenses have at least one surface that has a varyng radus of curvature, whch s desgned to mnmze sphercal aberraton. The aspherc surface mnmzes sphercal aberraton that would normally be present n a sphercal lens. Dffractve lenses usually consst of sphercal surfaces, one of whch has been etched or placed n contact wth a dffractng surface that reduces sphercal aberraton. Because of ease of fabrcaton, and therefore relatvely low cost, pano-convex s the most common type of lens used n materals processng, but may suffer from excessve sphercal aberraton at short focal lengths. Spot Sze and Depth of Focus nother mportant parameter n laser materals processng s the depth of focus of the laser beam. Because the absorpton of laser energy by the workpece s dependent on the focused laser spot sze (Ref. 3), t s very mportant to assure that the workpece surface s located wthn the regon of mnmum spot sze (.e., near z0). n practce, ths s accomplshed by usng the longest focal length lens that wll produce the requred depth of weld penetraton or cut qualty. Generally, depth of focus ncreases wth the focal length of the lens. practcal defnton has been gven by Marshall (Ref. 8). Marshall arbtrarly defned depth of focus as the dstance along the propagaton drecton at whch the power densty has decreased to 90% of ts value at the poston of the mnmum spot sze. Ths corresponds to an 11% ncrease n the area of the beam, whch occurs when!! -~t tl, n l! ll!!j l- ltan '. '{{::'::::'':::':T '.'. '.'. '.'." '. ".'.' ".','. '." ".'.'," '.' :!, : -~,,~1 l~,l Fg ntensty dstrbuton of a CW CO 2 laser operatng at 300 W. -- Contour and cross-sectonal dsplays; B -- sometrc dsplay. The laser radaton s focused by a 3.75-n. aspherc lens. The contour s taken at the plane, whch corresponds to 86.5% of the total ncluded power. ( z ) = 0.9 Rewrtng Equaton 8b as,2],,-,o) ] and substtutng w2(z)= 0.9 gves 0 F ] (9) (10) Solvng for the propagaton dstance z gves w0 z = (11a) The depth of focus (c3) s the dstance wthn whch the area (z) s related to the area at the mnmum wast, 0, by (z)_< 0.9 The length of ths regon s gven by = 2,~ 3M2,;L (11 b) Measurement of w 0, M 2, and therefore allows for a determnaton of the precson requred for the postonng devces used n materals processng. n ths paper we present a method of determnng beam qualty, mnmum spot sze and ts poston, and depth of focus by performng measurements of beam spot sze along the laser's propagaton drecton and then usng a nonlnear curve fttng routne to obtan an expresson relatng the spot sze of the laser beam to ts propagaton dstance. ro- tatng aperture s used to make all measurements on a 1200-W contnuous wave (CW) CO 2 laser. We present data to establsh the relatve performance of pano-convex, menscus, aspherc, and dffractve lenses. dscusson of lens aberratons and ther effects on beam qualty s also presented. Experment Dagnostc measurements were performed on a Photon Sources V kw CW CO 2 laser. Measurements of the focused and unfocused spot sze were made usng a rotatng aperture system (Prometec Laserscope) -- Fg. 2. Laser radaton s detected by means of a pyroelectrc detector mounted above a rotatable arm. The end of the arm has an aperture through whch the laser radaton passes. The radaton s reflected down the length of the arm and then reflected to the detector. The entre assembly translates along the y axs and rotates about the z axs, so that the ntensty dstrbuton of the unfocused or focused beam s measured. Sgnals from the detector are sent to a computer, and stored on dsk. The Laserscope therefore measures the cross-sectonal area of the beam at a pont along the z axs. Measurements are made ndependent of the laser mode, so that the technque may be used to perform beam radus measurements on beams consstng of hgher-order modes. ZnSe focusng lens s mounted on a lnear translaton stage and s accurately postoned along the laser propagaton axs. The Laserscope s mounted on an x-y translaton stage to facltate postonng ~ ll~,. ~[ ~ 1.1,11 ~.,U~, ~ Jl,l ' lld n Bw, ~:f): C~:: [ lull;:

4 50-s FEBRURY ~ ~ ~ m ' ' ',--~ 2 :.:::c:... o~o POSTON (mm) Fg Beam radus vs. lens poston for a CW CO 2 laser focused by 5-n. menscus and pano-convex lenses. of the focused beam onto the aperture. Fgure 3 shows a typcal measurement of spot sze. The Laserscope has been used to map the three-dmensonal ntensty dstrbuton of the laser and determne the area of the beam at the plane above whch 86% of the total power s ncluded. The spot sze radus s then taken to be the radus of a crcle havng the same area. The frst fgure s an sometrc dsplay of the ntensty of the laser n a plane perpendcular to the drecton of propagaton of the laser,.e., n the x- y plane. Ths dsplay type was used to vew the overall shape of the beam and also ndcated f the detector was saturated. The contour and cross-sectonal dsplays facltated algnment of the beam wth the aperture and also ndcated saturaton. The contour plot represents the cross-secton of the three-dmensonal sometrc ntensty dstrbuton taken at the plane correspondng to 86% of the ncluded power. The x and y secton plots represent crosssectonal vews of the dstrbuton and are taken n the x-z and y-z planes, re- Table 1 -- Lenses Tested spectvely. Measurements were made on 12 ZnSe lenses wth nomnal focal lengths rangng from 2.5 to n. (63.5 to 522 mm). lst of the lenses used s gven n Table 1. ll measurements were made usng approxmately 300 W of laser power. To begn measurements, the lens was typcally postoned so that the aperture of the Laserscope was a few mllmeters 13 o above z 0. Determnaton of z 0 requred teratve tral measurements, snce nne of the 12 lenses focused the laser radaton at a poston that was closer to the lens than the nomnal focal length value. 20-tJm aperture was used to take all measurements of beam radus. Measurements were made on both sdes of z 0 along the propagaton drecton, movng the lens n ncrements of 0.25 to 7 mm, dependng on the focal length of the lens. There are two major sources of error assocated wth measurements of spot radus. The frst s due to temporal fluctuatons n the spot radus of the unfocused beam, the second may result from an ncorrect calculaton of spot radus by the measurement apparatus. The temporal fluctuatons of the laser were n effect averaged out by takng measurements along the propagaton axs at dfferent ponts n tme and plottng the best-ft curve through the ponts. The ncluded power technque s vald for beams that are rotatonally symmetrc and have no ponts of zero ntensty (e.g. cold spots whch occur at the center of the donutshaped dstrbuton of an unstable resonator). The Photon Sources laser exhbts no cold spots and any errors ntroduced as a result of a noncrcular n- Nomnal Focal Length Lens type Manufacturer 2.50 n. (63.5 mm) aspherc Laser Pwr. Optcs 2.50 n. (63.5 ram) menscus -Vl nc n. (63.5 mm) dffractve Coherent 2.50 n. (63.5 mm) pano-convex -V nc n. (95.2 mm) aspherc Laser Pwr. Optcs 3.75 n. (95.2 mm) menscus -Vl nc n. (95.2 mm) dffractve Coherent 5.0 n. (127 ram) menscus -Vl nc. 5.0 n. (127 mm) pano-convex -V nc. 7.5 n. (190.5 mm) pano-convex -Vl nc n. (254 mm) menscus -Vl nc n. (522 mm) menscus -Vl nc. tensty dstrbuton are also averaged out over the measurements taken along the propagaton drecton. Furthermore, ncorrect measurements of beam radus are also prevented by holdng the rato of the area of the beam to the area scanned by the Laserscope (the fll factor) and the senstvty of the nstrument wthn the ranges recommended by the manufacturer. The fll factor was held between values of 0.40 and 0.80, and the senstvty was also adjusted durng each set of measurements so that the nstrument was not saturated as the aperture was moved about z 0. Results and Dscusson Mnmum Spot Sze Fgures 4-6 show data sets of the lenses that typcally have utlty n materals processng applcatons (.e., f < 5.0 n., 127 mm). The data correspond to the spot sze measurements made usng a 20- pm-dameter Laserscope aperture. Measurements were taken approxmately 3 mm above and below z 0 n ncrements of 0.25 to 7 mm, dependng on the lens focal length. The beam radus was plotted as a functon of propagaton dstance and a nonlnear curve-fttng algorthm was then employed to ft to the data a curve of the form of Equaton 8b. The unfocused beam radus was found to be W 0 = 9.65 mm. Fgures 4-6 show that wth curve fttng t s possble to quckly descrbe the propagaton of the focused laser beam along ts propagaton axs usng Equaton 8b. Ths curve fttng analyss s smple to perform on expermental data and provdes a powerful method for analyzng laser spot sze measurements (Ref. 9). three-parameter curve ft (wth parameters w 0, M 2, and z 0) such as shown n Fgs. 4-6 was performed for all 12 of the lenses that were examned. The results are summarzed n Fgs. 7-9 and n Table 2. Fgure 7 shows the spot sze as a functon of focal length for the dfferent lenses examned. The sold lne shows the deal spot sze obtaned usng Equaton 2, assumng that the n. (522-mm) focal length menscus lens ntroduces no sphercal aberraton. Equaton 2 mples that, n the absence of sphercal aberraton, the product woo s conserved as the lens focal length s changed. Ths yelds an expresson for mnmum spot sze gven by w = w (2 ~) ~ (12) where w 0 (20.5) s the spot sze obtaned usng the n. focal length lens, f, s

5 spherc WE,DNG RFSFRCH q[ JPP FMFNT ~1.c dfl?actve ~" ffl =l ~ \' sphere l ~,lenscus :,~ " ~l,:' 2; POSTON (mm) :, > E < r : ' _. s :! " :, ], ;,, \\ jr POSTON (mm) { F<~.5 Beam radus of a CW CO, laser focused by.7.75-n. tbcal lenqth lenses of varous shapes. and menscus; B l an arbtrary focal length, and w 0 s the spot sze at f. We suspect the domnant mechansm responsble for the devaton of the meascred spot sze from the deal spot sze s sphercal aberraton. Other thrd-order aberratons such as coma, astgmatsm, and dstorton are neglgble when laser radaton s focused usng hgl~-qualty lenses (Ref. 10). t f = 2.5 n. (63.5 turn), the ablty to focus the laser s hghly dependent on the lens type. The pano-convex lens has the largest devaton from the deal spot sze, wth a measured radus, whch s 67% larger than the radus predcted for an aberraton-free lens. However, for the 2.5-n. menscus lens, the ncrease over the deal spot sze s less than 44%, and only 23% for the specally fabrcated 2.5- n. aspherc lens. The performance of the pano-convex lens relatve to the other lenses s expected snce the pano-convex lens s uncorrected for sphercal aberraton. By usng lenses that have been corrected for sphercal aberraton (aspherc, menscus, and dffractve), rather than the less expensve pano-convex lenses, a sgnfcant ncrease n performance s obtaned at short focal lengths. These results support the hypo thess that at short focal lengths, sphercal aberraton lmts the focused spot sze even when hgh-qualty laser lenses are used. t f = L75 n. (95.2 mm), the aspherc lens agan produced the smallest spot sze, wth a focused spot radus of 82 vm, whle the mensccls ard dffractve lens spot szes were larger at 86 and 98.lm, respectvely. t a laser power of 30(7 W, the spot sze attaned wth the f = 2.5 n. dffractve lens s 60% larger than the deal vah.le. Lke the aspherc lenses, the dffractve lenses are specally machned to elmnate sphercal aberraton and produce a small spot sze. We postulate that the reason nether the dffractve nor the aspherc lenses acheved the predcted mnmum spot sze s because the shape correcton used to fabrcate these lenses s ncorrect. The requred shape correcton s determned usng computer lens desgn software that requres boundary condtons that nclude the laser beam characterstcs. Snce these lenses are produced for all CO 2 lasers, t s lkely that the chosen shape correcton s not optmum for the laser examned here. t should also be noted that the relatve performance of the dffractve lens may m \, Of) spherc Menscus ' o ",,1 ', x, ',\ ", ~~ ~ r \ ~, L \,, t ' \ \ \ \ \ \ \\, '~ T x\ '\ \ \ ~" \ : 7 ~7 : POSTON (mm) POSTON (mm) F~. 6 Beam radus vs. lens poston tor a CW CO, laser tb~ used by a 2.5-n. tocal length lens. -- spherc and menscus lenses; B -- dffractve and plano-conw,x lenses.

6 52-s FEBRURY O spl~r V-'.... t~ a ~ 2o0 w =o o Menscus Plano-C... ~ u Dffractve deal ( qn[ 121) [ v 160 _= n- =f 120 1,1,1 Menscus Pano-Convex Dffractve ---- deal (eqn[12]l J.y _z =E 100 =_ 80 Z o j NOMNL FOCL LENGTH (mm) B 4O NOMNL FOCL LENGTH (ram) Fg Spot sze of a CW CO 2 laser as a functon of lens focal length; B -- boxed secton of Fg. 7. prove at hgher laser powers, as stated by the manufacturer, when thermal effects on the lens become apprecable. Fgure 7 shows that for all of the lenses that have utlty n materals processng usng ths laser (.e., f <- 5.0 n., 127 mm), the deal mnmum spot sze s not obtaned usng a sngle element lens, even f the lens s an aspherc or dffractve type. The only lenses shown to have nsgnfcant sphercal aberraton are those that focus the laser to spot szes too large to produce meltng (.e.f>5.0 n.)n most engneerng alloys. Therefore, t s reasonable to conclude that the performance of ths laser, and lkely many other materals processng lasers, can be sgnfcantly mproved wth lens desgn that uses the proper boundary condtons to elmnate sphercal aberraton. Beam Qualty Table 2 gves the measured beam qualty for each focusng lens. Note that measured beam qualty s nversely proportonal to the focal length of the lens. Table 2 -- Summary of the Focused Laser Parameters Ths varaton n beam-qualty s lkely due to sphercal aberraton, snce beam qualty s an ntrnsc property of a materals processng laser and cannot be mproved by the ntroducton of optcal elements. s was the case for mnmum spot radus, the best performance (beam qualty) at short focal lengths was provded by the aspherc lenses, wth beam qualty measurements of M 2 = 2.4 at f = 2.5 n., and M 2 = 2.3 at f = 3.75 n. For laser materals processng personnel, the utlty of knowng beam qualty s that one s able to estmate the spot sze attanable wth a gven lens. However, snce the measured beam qualty s not consstent among lenses of dfferng shape and focal length, due to sphercal aberraton, one must take nto account the effect of sphercal aberraton on beam qualty n order to estmate spot sze. Of course, the effect of sphercal aberraton on measured beam qualty wll depend on the type of focusng lens used. The menscus shape was the shape most commonly tested n ths study. curve was ft through the data ponts of Lens Type (z0) (mm) 8 (mm) w0 (pm) M 2 asphere menscus dffractve pano-convex asphere menscus dffractve menscus pano-convex pano-convex menscus menscus t the menscus lenses (Fg. 8) usng an equaton of the form M2 _ M2 +,:t - 0 f2 (1 3) where M02 s the ntrnsc, aberraton- free laser beam qualty, a s a constant that depends on the lens shape and the beam profle, and f s the focal length of the lens. The term af 2 represents the n- crease n M 2 due to sphercal aberraton. Ths expresson n combnaton wth Equaton 2 can be used to estmate mnmum spot sze for menscus lenses based on the beam qualty (M02) and the focal length of the focusng lens. Substtutng Equaton 13 nto Equaton 2 and usng 0 = W0f yelds an equaton of the form w = clm2f + ~- (14) where c 1 and c 2 are constants. The frst term of Equaton 14 represents the lnear dependence of spot sze on lens focal length n an aberraton-free lens. The second term represents the ncrease n laser spot sze due to sphercal aberraton. t long focal lengths, the sphercal aberraton term s neglgble, and as predcted by Equaton 12, the spot sze vares lnearly wth f. However, at short focal lengths, the aberraton term becomes nonneglgble and results n a nonlnear varaton of spot sze wth focal length -- Fg. 9. Therefore, the effect of the aberraton term s to ncrease the devaton of the measured spot sze from the deal aberraton-free spot sze as the lens focal length decreases. The smallest value of M 2 = 1.5 was measured usng the n. focal length menscus lens. ssumng that ths lens ntroduces no sphercal aberraton

7 d" -.,,., o ' ' ' ' T ~ M 2 = 1, f 2 \... J,,, NOMNL FOCL LENGTH (mm) Fg Beam qualty of a CW CO, laser as a functon of lens focal length. The data are for a 19.3-ram-dameter beam focused by menscus lenses of varous to(al lengths. (M02 >~ af2), the beam qualty of the laser s approxmately M02 = 1.5. The curve ft of Fg. 8 yelded a = 5.96 x 103. Ths gves af 2 ~ 0.02, so that at long focal lengths, the ncrease n beam qualty due to sphercal aberraton s on the order of the uncertanty n the measurement of beam qualty. Takng the uncertanty n M 2 to be the same as the uncertanty n a measurement of the beam radus (- 6%), we determne that the beam qualty of the laser s M02 = ].5 +_0.09. Fgure 9 shows the ft of the spot sze data for the menscus lenses obtaned usng Equaton 14. The ft yelded c t = 4.8 xl0 4 and c 2 = 3.52 x 10 ~ mm 2. From Fg. 9 t can be seen that Equaton 14 accurately predcts the effect of sphercal aberraton on spot sze at small focal lengths. The fgure also shows that the effect of sphercal aberraton becomes neg- lgble as the focal length ncreases, where the varaton of spot sze approaches the lnear dependence predcted by the aberraton-free model of Equaton 1 2. Poston of Mnmum Wast and Depth of Focus The three-parameter curve fttng analyss also readly deter- 600 mnes the poston (z 0) of the mnmum spot radus (w{~). The result- ng z 0 values are gven n Table 2. One can see that for most lenses the effectve focal length s somewhat shorter than the nomnal focal length. These results are consstent wth the effect of sphercal aberraton n geometrc optcs, where nonparaxal rays are brought to a focus nearer the lens than are paraxal rays (Ref. 7). n materals processng applcatons, the effectve shortenng of l e focal length of a lens results n less Jorkng dstance between the workpece and the lens. n addton, ths varaton n the poston of the mnmum spot radus also necesstates an emprcal determnaton of z 0 by the laser operator, snce use of the nomnal focal length of the lens s nexact. Depth of focus was also determned for each lens usng Equaton 11b. The resuts are gven n Table 2. Typcal values of depth of focus were approxmately 0.5 and 1.0% of the nomnal focal length for the 2.5- and 5.0- n. menscus lenses, respectvely. Note that as sphercal aberraton lmts the mnmum spot sze obtanable at a gven focal length, t results n an ncreased depth of focus. Ths effect s a demonstraton of the prncple of conservaton of rradance. When a laser beam s focused to a smaller spot, there s necessarly an ncrease n the dvergence, and therefore, a smaller depth of focus. The values determned for the depth of focus provde an upper lmt for the precson of the fxtures used to poston the beam relatve to the workpece. Depth of focus also ndcates the allowable dmensonal varaton of the parts to be pro(essed n the drecton perpendcular to the lens. deally, the precson should be no larger than half the value of the depth of focus of the lens. Substtutng w(~ = M2;Lx0 nto Equaton 11 b and usng 0 = Wf gves 6 = CM2t 2 (1 5) where C s a constant. Substtutng Equaton 113 nto Equaton 15 yelds 6 g'(','+'w:) ( ~) where c ' and c 2' are constants. By fttng a curve of the form of Equaton 16 to the menscus lens data, we determne c f' = 4.59 x 10 5 mm and c 2' = 58 mm 2. Therefore, usng Equaton 16, the depth of focus may now be determned for the laser used n ths study, focused by a menscus lens of any focal length. Conclusons 1 ) The laser propagaton equaton was ftted to spot sze measurements taken near the focal plane of a CW CO 2 laser for 1 2 dfferent focusng lenses, wth excellent agreement obtaned for each lens. 2) Measured values of spot radus and ,,...~ ' E B ll m z L 0 Me--red... predcled d ncludng aberralon ~ deal (eqnl 121} " \ w o = c 15r + c2f,x'a#" M~asu~d Predcted (ncludng aber~lon ) lde~d (~n[12])," ; 0-- w = cl 1.5f + cjf,,"" 120," m.'.-" ;;} ",",' NOMNL FOCL LENGTH (mm) B NOMNL FOCL LENGTH (ram) Fg Mnmum beam radus of a CW C(), laser as a functon of lens focal length tor fve menscus lenses; B boxed secton of Ft;. 9. The data ponts are ft wth a curve of the fom~ of Equaton 14. The constants are c = 4.8e "~ and c 2 = 3.52e ~. WELDNG RESERCH SUPPLEMENT 53-~

8 many of the lenses examned. These ncreases were judged to be due to sphercal aberraton by the focusng lens. 3) n general, lenses corrected for sphercal aberraton produced better results (smaller spot sze and mproved beam qualty) than the pano-convex lenses. t short focal lengths, the best results were obtaned usng aspherc lenses. 4) Due to the nonoptmum performance of the lenses tested, an mproved lens desgn that mnmzes sphercal aberraton s expected to reduce focused spot sze and thereby ncrease the processng capabltes of materals processng lasers. 5) The poston of the mnmum spot radus was determned for the lenses examned, and found to be nearer to the lens than the nomnal focal length value. 6) Depth of focus was defned and calculated for the lenses examned n ths work, and was found to be dependent on the beam qualty of the laser and on the amount of sphercal aberraton produced by the focusng lens. 7) Equatons have been determned for the menscus lenses tested, whch permt calculaton of the mnmum spot sze and the depth of focus based on knowledge of the laser beam qualty and the focal length of the lens. 8) The beam qualty of a 1-kW CW CO 2 laser was determned to be M 2 = 1.5 _ Ths measurement was made usng a long focal length lens (f = n., 522 mm), so that contrbuton to M 2 from the lens due to sphercal aberraton was comparable to the uncertanty n a measurement of M 2. cknowledgments The authors would lke to acknowledge Fred Hooper for hs techncal assstance. We also would lke to thank Charle Robno for a careful revew of the manuscrpt. Ths work performed at Sanda Natonal Laboratores supported by the U.S. Dept. of Energy under contract number DE-C04-94L References 1. Prokhorov,. M., Batanov, V.., Bunkn, F. V., and Fedorov, V. B Metal evaporaton under powerful optcal radaton. EEE Journal of Quantum Electroncs QE- 9(5): Herzger, G., and Kreutz, E. W Fundamentals of laser mcroprocessng of metals. Physca Scrpta T13: Fuerschbach, P. W Meltng effcency n fuson weldng. The Metal Scence of Jonng. TMS, Cncnnat, Oho, pp Jones, R. D., and Scott, T. R Laser beam analyss pnponts crtcal parameters. Laser Focus World, January, fflander, R Focusng of multmode laser beams wth varable beam parameters. Optca cta 33(8): Kogelnk, H., and L, T Laser beams and resonators. ppled Optcs 5(10): Smth, W. J Modern Optcal Engneerng. Second ed., McGraw-Hll, New York, p Marshall, H. L pplcatons a la mode. Laser Focus, prl, pp Fuerschbach, P. W., and Essen, M Beam qualty measurements for materals processng lasers and the proposed SO standard. CLEO, L, Orlando, Fla. 10. Roberts, D Laser optcs: desgn consderatons. The Photoncs Desgn and pplcatons Handbook, pp sl FEBRURY 1996