Theory of Geometrical Methods for Design of Laser Beam Shaping Systems

Transcription

1 Heade fo SPIE use Theo of Geometical Methods fo Design of Lase Beam Shaping Sstems David L. Sheal Univesit of Alabama at Bimingham Depatment of Phsics, 153 3d Avenue South, CH31 Bimingham, AL USA ABSTRACT Fom geometical optics, the laws of eflection and efaction, a tacing techniques, consevation of eneg within a bundle of as, and the condition of constant optical path length povide a foundation fo design of lase beam shaping sstems. Geometical optics design methods ae pesented fo shaping the iadiance pofile of both otationall and ectangula smmetic lase beams. Applications of these techniques to design eflective and efactive lase beam shaping sstems ae pesented. Kewods: lase; beaming shaping; geometical optics; optical design; iadiance mapping 1. INTRODUCTION Geometical optics is used to shape the iadiance pofile as pat of the optical design of lase beam shaping sstems. The laws of eflection and/o efaction, a tacing, consevation of eneg within a bundle of as, and constant optical path length condition ae used to design lase beam pofile shaping optical sstems. Intefeence o diffaction effects ae not consideed as pat of the design pocess. Onl lenses and mios ae used fo the optical components of the beam shaping sstems pesented in this pape. Refeence 1 also descibes how to use geometical optics to design lase beam shaping sstems. Thee ae man divese applications of lases in science and technolog. These applications use a vaiet of the unique popeties of lases, such as, the high intensit, coheent, monochomatic light of lases. Fo illumination applications, it is impotant fo the lase beam to unifoml illuminate the taget suface. Tuncating a Gaussian beam with an apetue is a wa to moe unifoml illuminate the taget suface with a lase beam. Howeve, intensit apodization techniques ae inefficient. Both eflective 3, 4, 5 and efactive 6, 7, 8 optical sstems have been used to shape lase beam pofiles. McDemit and Hoton 3 use consevation of eneg within a bundle of as to design otationall smmetic eflective optical sstems fo illuminating a eceive suface in a pescibed manne using a non-unifom input beam pofile. Malak 5 has designed a two-mio lase pofile shaping sstem whee the second mio is decenteed elative to the fist 9, 1 mio to eliminate the cental obscuation pesent in the aiall smmetic design fo eflective sstems. Conwell pesents geneal esults that can be used to design lase beam shaping sstems with otational, ectangula, o pola smmet about the optical ais. Keuze 6 has patented a coheent-light optical sstem using two aspheical sufaces to ield an output beam of desied intensit distibution and wavefont shape. Rhodes and Sheal 7 deived a set of diffeential equations using intensit mapping to calculate the shape of two-aspheical sufaces of a lens sstem that epands and convets a Gaussian lase beam pofile into a collimated, unifom iadiance output beam. Using thei method, two-plano-aspheical lenses have been 11, 1, 13 designed, fabicated and used fo lase beam shaping in a hologaphic pojection sstem. A theo fo designing the optics of a lase beam shaping sstem is pesented in section. A bief oveview of the concepts of as, wavefonts, and eneg popagation within geometical optics is pesented. Then, specific attention is devoted to using consevation of eneg within a bundle of as, a tacing, and the constant optical path length condition as constaints on the optics figue of lase beam shaping sstems. Section 3 pesents thee applications of this theo of lase beam shaping using one-mio, two-lens, and two-mio configuations. Othe autho infomation: dls@uab.edu ; Telephone: ; Fa: Copight Societ of Photo-Optical Instumentation Enginees. This pape will be published in Poc. SPIE 495 and is made available as an electonic pepint with pemission of SPIE. One pint o electonic cop ma be made fo pesonal use onl. Sstematic o multiple epoduction, distibution of multiple locations via electonic o othe means, duplication of an mateial in this pape fo a fee o fo commecial puposes, o modification of the content of the pape ae pohibited.

2 . THEORY OF LASER BEAM SHAPING.1. Oveview of the Concepts of Ras, Wavefonts, and Eneg Popagation within Geometical Optics 14, 15 The concepts of as, wavefont, and eneg popagation ae fundamental to undestanding and using geometical optics to design a lase beam shaping sstem. In ode to detemine o optimize the iadiance within an optical sstem, the optical field must be detemined thoughout the sstem. The optical field is a local plane wave solution of Mawell's equations o 16, 17 the scala wave equation. Fo an isotopic, non-conducting, chage-fee medium, the optical field ma be witten as: u u ep ik S( ) (1) whee u epesents the components of the electic field at an point, n is the inde of efaction at, k ω c π λ is the wave numbe in fee space, ω is the fequenc of the wave, c is the speed of light, and λ is the wavelength of light. u and S( ) ae unknown functions of. Requiing u() fom Eq. (1) satisf the scala wave equation leads to the following conditions which must be satisfied b u () and S( ): b S g n () u S u + u S (3) whee the tem popotional to c1 k h has been neglected. Equation () is known as the eikonal equation and is a basic equation of geometical optics. The sufaces S (,, z) const. ae constant phase fonts of the optical field, have a constant optical path length ( OPL) fom the souce o efeence suface, and ae known as the geometical wavefonts. Fo isotopic media, as ae nomal to the wavefont. A unit vecto nomal to the wavefont and along a a at the point is given b a S S S. n Equation (3) is equivalent to consevation of adiant eneg within a bundle of as and leads to the geometical optics intensit law fo popagation of a bundle of as. Using the vecto identit ( f v) f v + v f, Eq. (3) can be c h c h ewitten as u S u na. Then, ecognizing that the eneg densit of a field is popotional to the squae of the field amplitude u o and that the intensit I is equal to the eneg densit of the field times the speed of popagation within medium, Eq. (3) can be witten as ( I ) a. Fo beam shaping sstems with collimated input and output beams as illustated in Figue 1, a useful epession fo the eneg within a bundle of as 18 as it passes though the sstem follows b integating Eq. (5) ove efeence planes (o wavefonts) nomal to the input and output beam and then appling Gauss theoem I dw I dw. in out Equation (6) epesses consevation of eneg along a bundle of as between input element of aea dw π d and the coesponding output element of aea dw ( π Rd) on the wavefont o the efeence planes nomal to beam. Equation (6) is a basic equation used to design lase beam shaping sstems. (4) (5) (6) Fo some beam shaping configuations, it is necessa to intoduce the consevation of eneg condition into the optical design b using efeence sufaces, such as fo detectos, which ae cuved and/o have an abita oientation with espect to the diection of the beam popagation. In these cases it is necessa to take into account pojecting the element of

3 aea of a efeence suface pependicula to diection of beam popagation when appling Eq. (6). Seveal eamples will be pesented late in this pape to illustate cases using Eq. (6) as pat of the design of lase beam shaping sstems. Accoding to geometical optics, the phase and amplitude of the optical field ae evaluated independentl. Fist, the a paths ae evaluated thoughout the optical sstem, which enables computing the phase in tems of the optical path length of as passing thought the sstem. The amplitude of the optical field is detemined b monitoing the intensit vaiations 19,, 1 along each a. Figue 1. Schematic laout of a lase beam pofile shaping sstem. Ras geneall chaacteize the diection of popagation of adiant eneg, ecept nea foci o the edge of a shadow whee intefeence and diffaction takes place. As such, a a is a mathematical constuct athe than a phsical quantit. Snell's law elates the diection of incident and efacted as at an inteface between media of diffeent indices of efaction, which can be witten in vecto fom n A na+ n cosi ncosi n (7) whee a and A ae unit vectos along the incident and efacted as, n is a unit vecto along the nomal to the inteface suface with the geneal oientation of the incident a, ( nn, ) ae the indices of efaction of the incident and efacting media, (ii, ) ae the angles of incidence and efaction, cos i An ˆ, and cos i an ˆ. When mios ae involved, Eq. (7) can be used to compute the diection of the eflected a b setting n n and using the optics sign convention. 3 Eplicitl, a unit vecto A along a eflected a is given b A a n(a ˆ n ˆ). Beam popagation is commonl descibed b wavefonts. A wavefont is a suface of constant phase of the wave o optical path length (OPL) fom the souce o efeence suface. Each a geneall follows the path of shotest time though the optical sstem accoding to Femat's Pinciple which states that a a fom points P to Q is the cuve C connecting these two points such that the integal OPL( C) n(,, z)ds C is an etemum (maimum, minimum, o stationa). The quantit nz (,, ) is the inde of efaction of the medium, and ds is the infinitesimal ac-length of the cuve. Fo a homogeneous medium, the optical path length between P and Q is the geometical path length between the two points times the inde of efaction of the medium. In geneal, the optical path length divided b the speed of light in fee space, c, gives the time fo light to tavel fom point P to Q along the a path C. 4 The a path C can be detemined using the calculus of vaiations. When the inde of efaction, n, is a smooth 17 function, it can be shown that the a path C satisfies the diffeential equation (8) (9)

4 F I HG K J (1) d ds n d n ds whee is the position vecto of an point on the a. Equation (1) is known as the a equation and is difficult to solve in man cases. Fo homogeneous medium ( n const ), the a path is epesented b a staight line ( s) as+b whee a and b ae constant vectos, and s is the a path length. Constant inde of efaction mateials ae used in man optical sstems. When the inde of efaction is a function of position, such as, the aial distance, z, the a paths ae cuved... Consevation of Eneg Condition An oveview of using the consevation of eneg condition to design lase beam shaping sstems is pesented in this section. Fist, sstems in the configuation of Figue 1 ae consideed. It is shown how to elate the input and output beam diametes to conseve eneg fo a Gaussian input beam. Then, it is shown how to use consevation of eneg, Eq. (6), to design beam shaping optics when a non-collimated output beam is incident upon a cuved detecto...1. Collimated Input and Output Beams Assume the incident beam is a lase in the fundamental, Gaussian TEM mode with cental intensit nomalized to unit. Then, I (1) in ep[ ( ) ] whee 5 is the adius of the beam. The conventional definition of the diamete of a lase beam o waist is the location at 1 e e which the field amplitude is of its peak value. At the beam waist the beam intensit is of its aial value. Fo a Gaussian beam, 86.5% of the total eneg of the beam is contained within the waist. This beam leaves the optical sstem at a adial distance R fom the optical ais with a powe densit of I ( R ) which is commonl sought to be constant as a esult of the beam shaping design pocess. Integating Eq. (6) ove the input and output planes gives π π dθ I () d dθ Iout ( R) R dr (13) in R whee eflection and absoption losses have not been consideed. When I ( R) is constant, the adius of beam in the eit apetue R can be evaluated b caing out the integation using Eq. (1) fo Iin() to obtain out out (11) R 1 ep I out ( ) (14) whee I 1 ep( ). (15) out Rma ma In Eq. (15), ma is the woking input apetue of the optical sstem, and is the coesponding point on the output plane. Equation (14) is used duing the optical design pocess to educe the numbe of independent vaiables and detemine the shape of the eflecting o efacting sufaces of the beam shaping sstem. Equation (14) elates the output beam adius to that of input beam, R, fo sstem configuations illustated in Figue 1. R ma

5 ... Unifom Iadiance ove Cuved Detecto When a beam shaping sstem does not have eithe a collimated input o output beam as illustated in Figue 1 o the output efeence suface is cuved, the consevation of eneg condition, Eq. (6), needs to be evised to incopoate the actual geomet. In these cases, it is geneall not possible to integate the esulting consevation of eneg condition [evised Eq. (6)] and then to solve fo R() as done in Eq. (14). Rathe, the optical design of these tpes of beam shaping sstems can be accomplished b incopoating the mapping ( z, ) RZ, of the a tace equations into the epession of b consevation of eneg fo the beam shaping sstem, and then solve the esulting diffeential equation fo the contou of the eflecting o efacting sufaces. This pocedue is illustated in section 3.1 fo a one-mio beam shaping sstem..3. Ra Tace Equations Fo the efactive beam shaping sstem shown in Figue, as ae efacted at suface s accoding to Snell's law. The diection of efacted a A taveling fom the point ( z, ) on suface s to the point ( RZ, ) on suface S is given b Eq. (7) whee the incident as ae along the optical ais, that is, a k. Eplicitl, a unit vecto along the efacted as ove suface s is given b A γ k + Ω n (16) whee γ nn ; z dz() d ; and o bgc b g Ω γ z 1 γ 1+ z 1 d i b g ht 1 g, (17) n z + k 1+ z 1. (18) The unit vectos (, k ) ae along the z diections. Accoding to Eq. (11), the a path connecting (, z ) and ( RZ, ) is a staight line with slope given b Eq. (16) and can be epesented b ( R )( A) ( Z z)( A) (19) z whee ( A) and ae the (, z) components of A given b Eq. (16). Combining Eqs. (16) and (19) ields, afte A z squaing, collecting tems in powes of z, and simplifing: c hb g b g b g b gb g b g 1 γ Z z γ R z + R Z z z + R. () In Eq. (), R is given b Eq. (14), and z is the solution to the diffeential equation as a function of the entance apetue coodinate. Z in Eq. () is not et known. Fo beam shaping applications, which seek to illuminate a specific suface S with unifom intensit, the equation of the suface S epessed as Z Z( R) will be adequate to solve the diffeential equation () fo the shape of the suface s. Fo othe beam shaping applications, which seek to also contol the shape of the output wavefont, the constant optical path length condition fo as passing though the sstem is used to detemine the shapes of the sufaces s and S, when solved simultaneousl with Eq. ().

6 Figue. Geometical configuation of a two-lens lase epande. (Fom Ref. 11.).4. Constant Optical Path Length Condition In ode fo the output wavefont to have the same shape as the input plane wavefont, it is necessa fo all as passing though the optical sstem to have the same optical path length. Fo unifom inde media, the optical path length of a a is the sum of the geometical path length of a specific a times the inde of efaction of the component of the sstem. Fo the two-lens beam shaping sstem shown in Figue, the optical path length ( OPL) of a a passing along the optical ais is ( OPL) nt + n d + nt. (1) 1 Fo an abita a of height, the optical path length is b g b g 1 b 1 g. () ( OPL) nz + n R + Z z + n t + d + t Z whee the oigin of the z-ais emains fied at the vete of the fist optical suface in these consideations. The constant optical path length condition is satisfied fo this sstem b equiing ( OPL) ( OPL). Combining Eqs. (1) - (3) enables one to epess ( Z z) in tems of when using Eq. (14). Thus, the constant optical path length condition helps to detemine the sag of one optical suface of a beam shaping sstem..5. Optical Design of Lase Beam Shaping Sstems Optical design of a lase beam shaping sstem seeks to define the optical components adequatel so that the sstem can be analzed, fabicated, and tested. This geneall equies specification of the shapes of and spacing between the optical sufaces as well as the inde of efaction of all the media. Fo the efacting beam epande sstem illustated in Figue, the shape of sufaces s and S must be detemined. The consevation of eneg condition and the constant optical path length condition can be solved simultaneousl with the a tace equations fo R, z, Z when nd,, t1, t ae given. In contast to conventional optical design, the pesent method of solving diffeential equations defines the optical sufaces s and S b poducing tables of numeical data (, z) fo suface s and ( RZ), fo suface S. It does not seem possible to 5, 9, 1 solve these diffeential equation analticall fo z and ZR b g. Howeve, it is inteesting to note that seveal authos have shown that the shape of the fist element of a beam shaping sstem can be epessed as a function of (3)

7 z z f( d ) + C (4) whee C is a constant, and f is a function elating to the optical configuation. The shape of the second suface can be computed fom the following epession Z z + g (5) whee g is anothe function fom the configuation. Section 3 descibes how to design a one-mio, two-lens, and twomio beam shaping sstems that can unifoml illuminate a detecto with an input lase beam and epand o magnif the beam diamete. 3. Optical Design of Lase Beam Shaping Sstems The optical design of lase beam shaping sstems involves incopoating the geometical optics intensit law fo popagation a bundle of as (consevation of eneg) and the constant optical path length condition into the a tace equations fo the optical sstem and then detemining the geometical shapes of seveal optical sufaces (o GRIN mateials) so that the beam shaping design conditions ae satisfied. Fo beam shaping applications that onl need to illuminate a given suface with a specific iadiance distibution, the constant optical path length condition will not be pat of the design pocess One-Mio Pofile Shaping Sstem Conside the otationall smmetic geomet of a one-mio beam shaping sstem shown in Figue 3. The adiation is incident upon the mio suface s with equation z z(). The equation of the eceive (detecto) suface S is Z ( R). The iadiance of the incident beam, Iin(), is incident upon a cicula ing about the z-ais of aea da π d and is eflected to a cicula ing on the eceive suface S of aea da π R 1+ dz dr dr. The input adiation is collimated (paallel to optical ais) with a known intensit pofile. It is desied to iadiate the eceiving suface (detecto) with a pescibed intensit distibution without placing an conditions on the shape of the wavefont of the output beam. Unit vectos along the input and output beams ae given b a k ˆ ( z ) ( 1+ z ) z ˆ 1 A a nˆ a nˆ kˆ (6). (7) The a tace equation connecting the mio suface s with the eceiving suface S in the -z plane is given b R A z ZR z Az 1 z Detecto, S Z( R ) b g b g. (8) a Mio, s Figue 3. Geometical configuation of a one-mio beam pofile shaping sstem. Z

8 Equation (8) can be witten as b g b g b g R z + Z z z + R. (9) Appling the diffeential eneg balance Eq. (6) to this poblem gives 1 dz Iin πd Iout( R) πr dr + dz Iout( R)πR 1+ dr. (3) dr Displaing the tem ( dr d) in Eq. (3) gives 1 dr Iin d Iout R R dz 1+ dr Recall that (),,and in out. (31) I I R Z R ae known functions of thei espective vaiables, and z is an unknown function at this point of the analsis. Also, note that the a tace Eq. (8) epesses a mapping between sufaces s and S: b (, z) RZ, g which implies that R is a function of, b g R. B implicit diffeentiation of Eq. (9) with espect to, it will be shown below how to incopoate the diffeential eneg balance equation (31) into the a tace equation fo this poblem, and theefoe, obtain a diffeential equation fo the unknown shape of the mio suface s. Diffeentiating Eq. (9) with espect to gives F I HG K J LdZ O F + + NM QP + dr b g b g I z HG 1 K J. (33) d d b g in Eq. (33) can be witten as dr zz R z 1 z Z z z d Fom the chain ule fo diffeentiation of a function of a function, the tem dz d dz d dz dr Z ( R) dr dr d d whee Z ( R) dzbrg dr can be evaluated diectl fom the equation of the suface S. Combining Eqs. (34) and (33) leads to dr z z ( R ) + ( Z z) + 1 z + zz ( + z ) d 1. (35) Rewiting Eq. (9) in the fom R bz zg b g z 1, z enables Eq. (35) to be witten as z z dr d. (37) ( R ) ( 1 z ) 1 z zz ( 1 z ) The tem dr d can be eliminated between Eqs. (31) and (37), and then the following diffeential equation is used to detemine the sag z : b g (3) (34) (36)

9 1 z z dz 1 I 1 z + z ( 1 z ) dr in (38) z ( R ) Iout ( R) R dz 1+ dr Equation (38) is equivalent to Eq. 13 of Ref. 3 and Eq of Ref. 4. When appopiate bounda conditions ae given, Eq. I R (38) can be solved fo the shape of the mio used to illuminate the eceive suface S with a pescibed intensit fo a given souce intensit pofile I (). Refeences 3 and 4 develop an etension of this analsis to two-mio intensit in pofile shaping sstems. A numbe of specific solutions fo both one- and two-mio sstems ae given in Refs. 3 and 4 including two lase beam pofile shaping sstems, such as, unifom illumination of a plane pependicula to the incident beam using a one-mio sstem fo an input Gaussian beam, Fig. 7.3 of Ref. 4, and unifom illumination of a plane on the optical ais with a two-mio sstem fo an input Gaussian beam, Fig. 7.9 of Ref Two-Lens Beam Shaping Sstem Fo optical sstems when the sag of two sufaces can be used in the optical design pocess, it is possible to shape the output beam intensit pofile and to keep the optical path length constant (etain the wavefont shape) fo all as passing though the sstem. Futhe, it is often desiable to epand the lase beam diamete. To achieve these design goals, the optical suface shapes o inde of efaction pofiles ma be used as design vaiables. Conside the geometical configuation of a efacting lase beam pofile shaping sstem shown in Figue. Fo applications using monochomatic lase beams, thee ae no chomatic abeations pesent, and it is satisfacto to use the same mateial (inde of efaction) fo all lenses. The two cuved sufaces (eflecting o efacting, depending on the application) ae used to satisf the design conditions fo the beam shaping sstem. Ras ae efacted at suface s accoding to Snell's law. The diection of efacted a A taveling fom the point (, z ) on suface s to the point ( RZ, ) on suface S is given b Eq. (16). The a path connecting ( z, ) and ( RZ, ) is a staight line accoding to Eq. (11) and is eplicitl given b Eq. (19) which can be eaanged as a quadatic equation fo z, Eq. (). Solving Eq. () as a quadatic equation fo z gives ( R )( Z z) ( n n )( R ) ( Z z) ( R ) (1 ( nn) ) ( Z z) ( nn) ( R ) ± + z whee the positive solution fo z is used fo the lase shaping lens configuation shown in Figue so that the fist lens is divegent. Fo this lens sstem, the constant iadiance of the output beam is computed fom Eq. (15); the height of a a R at the second lens fo each a in the entance pupil with height is computed fom Eq. (14). b Z zg is detemine b the constant optical path length condition, Eqs. (1) - (3), o eplicitl 1 b g b g b g b g, (4) n R + Z z n Z z d n n which can be viewed as a quadatic equation fo ( Z z) as a function of entance pupil apetue adius. Afte squaing Eq. (4) and collecting tems, the solution of the esulting quadatic equation fo ( Z z) is nn ( n) d+ n( n 1) d + ( n n )( R ) ( Z z) n n 1 whee the positive sign of the adical has been used so that the solution educes to the appopiate value of ( Z z) d when R. (39) (41) out

10 It is inteesting to note that combining Eqs. (39) and (41) pemits z to be epessed as a function of, thus, enabling z to be evaluated b an integation, as illustated in Eq. (4). Then, the shape of S is computed fom Eq. (41), 9, 1 which has the functional fom of Eq. (5). Conwell and Malak 5 have eached a simila conclusion. Eplicitl fo the two-lens beam shaping sstem shown in Figue, it follows fom Eqs. (14) and (41) ( Z z) g () nn ( n) d+ n( n 1) d + ( n n) 1 ep I ou t n n Similal, compaing Eqs. (4) and (39) leads to the following epession fo f ( ) b b ac f ± 4 a whee ( ) ( 1. (4) a () 1 nn g nn c ), (44) (43) b c g, (45) c () ( R ) 1 ep. (46) I out Then, z is detemined b numeicall integating z fom Eq. (4) o (39), and Z is evaluated fom Eq. (5) o (41). A FORTRAN pogam fo computing ( z, ) and ( RZ, ) is given in Appendi A of Ref. 1. Solving the diffeential equations using these pogams defines the optical sufaces s and S b poducing tables of numeical data (, z) fo suface s and ( RZ), fo suface S. To analze the pefomance of these lase beam shaping optical sstems with conventional optical design packages (CODE V 6, OSLO 7, o ZEMAX 8 ), the data epesenting the sufaces s and S has been fit to the conventional optics suface equation 9 of a conic tem plus otationall smmetic aspheical defomation tems fo the lase shaping sstem. The optics suface equation can be witten in the c z κ c b g N i 1 A i i (47) A i whee c(vete cuvatue), κ (conic constant), and (coefficients of the polnomial defomation tems) ae suface paametes that ae detemined b the fitting pocess fo each suface. A nonlinea least squaes fitting pogam based on the simple method 3 has been successfull used 11, 1, 13 to epesent the optical sufaces of a lase pofile shaping sstem. The flu flow equation has been used to compute the iadiance along a a as it popagates though the optical sstem. 7, 8 Altenativel, adiometic calculations fo optical sstems can be pefomed b using unit sphee method 19 o b witing a maco fo optical design softwae package to compute the iadiance ove an output suface of the lase shaping sstem. 31 Refeence 3 pesents esults fo design, fabication, and testing of a two-lens lase beam shaping sstem in the configuation shown in Figue Two-Mio Beam Shaping Sstem As the final application of the beam shaping theo developed in section, a two-mio lase beam shaping sstem with ectangula smmet will be descibed. Fo moe details and applications of two-mio beam shaping optics, the inteested

11 eade is efeed to Refs. 1, 5, and 9. Conside the two-mio beam shaping sstem illustated in Figue 4. The input and output beams ae collimated and paallel to the optical ais. Assume the input beam iadiance is given b Iin(, ) ep ep whee (, ) ae the beam waist in the, diections, espectivel, and the cental intensit is nomalized to unit. The output beam iadiance is unifom and is given b I A A const. out X Y (48) (49) Fo convenience, a scaling paamete fo linea (tansvese) dimensions is intoduced as follows: ; ma ; 3 ; ma 3 ; X Y. ma ma (5) Figue 4. Geometical configuation of a two-mio lase beam shaping sstem with ectangula smmet. The elationship between the element of aeas on the input and output planes can be witten in the following fom in (, ) I dd I dxdy. (51) out The total eneg must also be conseved which is epesented b integating Eq. (51) ove the full apetue of the input and output planes

12 ma ma ma ma X Y Ein ep d ep d AX dx A Y dy ma. (5) ma Xma Y ma o eplicitl afte evaluating the integals π 3 E ef ef 16I 4 4 π out in fom which it follows that Iout 3π ef (54) 18 Fo inteesting applications, it is desiable to have a non-unifom shaping of the lase beam pofile in othogonal diections. Theefoe, assume that thee is an independent and non-unifom magnification of the and a coodinates between the input and output planes: X m ( ), (55) Y mb g. (56) The ectangula magnifications m m and b g (53) can be detemined b imposing the incemental epession of consevation of eneg, Eq. (51), fo the sepaated intensit functions, Eqs. (48) and (49). Using appoach poposed b Ref. 9, the output iadiance epessions along the, diections ae given b A A X Y 1 16 π ep d ( ) π ep d ( ) and the output plane position coodinates X, Y ae given b ef 1 u X ep du A X ef ( ) ef 1 3 v Y ep dv A 3 Y ef ( ) ef, (57) ef, (58) Equations (59) and (6) can be used to evaluate numeical tables elating (, ) to (, ) numeicall the magnification equations (55) and (56) so that one can epess (, ) in tems of (, ) Z ( XY, ) in Eq. (78) below. (59) (6) X Y, which can then be used to invet X Y when evaluating

13 Now, the a tace equations connecting points on the input plane to the output plane ae developed. The geometical configuation of a two-mio lase beam pofile shaping sstem is illustated in Figue 4. The unit a vecto A 1 connecting the two mio sufaces s [, z, (, )] and S[ X, Y, Z( X, Y)] can be witten in the following fom: bx g i + by hg j+ bl + Z zgk A1 (61) b X g + by g + bl + Z zg whee L is the distance along the z-ais sepaating the local coodinate sstem on each mio and h is the distance along the -ais sepaating the local coodinate sstem on each mio. The functions z (, ) and ZX,Y ae the sag of each mio in the local coodinate sstem of each mio. Using Eq. (8), the a vecto connecting each mio ma also be witten in tems of the slope of the fist mio: z ˆ ˆ 1 ˆ i zj + z z k A ˆ( ˆ 1 a n a n ) (6) 1 + z + z whee a k, (63) i j n z + z k, (64) 1+ z + z z b g b g z z, z, and. (65) Equating the - and -z components of the a vecto A 1 fom Eqs. (61) and (6) gives the following a path equations z z X Y h, (66) z 1 z z. (67) X L + Z z Solving Eqs. (66) and (67) z z b gdc b g X L + Z X, Y z(, b X g + by hg ± leads to a quadatic equation with the following solutions hi b g dc b g hi b g b g X L + Z X, Y z(, + X + Y h b X g + by hg. (68) b g Zb X Yg Equation (68) is a patial diffeential equation fo the unknown mio suface functions z, and,. The constant optical path length b OPLg condition povides anothe independent condition to be satisfied b the mio suface functions z b, g and Zb X, Yg. The OPL of an aial a and geneal a ae given b L +, (69) OPL Aial Ra h l

14 ( OLP X + Y h + L + Z X, Y z(, ) + Z( X, Y ) z, ). (7) Geneal Ra Howeve, the OPL is constant fo all as. Equating the ight-hand-side of Eqs. (69) and (7) gives the following: ( ) ( l Z( X, Y) z, X + Y h + L + Z X, Y z(, ). (71) Squaing Eq. (71) leads to ( X) ( Y h) ( L+ l ) + + h Z X, Y z(, ). (7) b g b g Using the negative sign in Eq. (68) as the phsicall meaningful solution, it has been shown 5 that combining Eqs. (7), (68), and (66) leads to the following epessions fo z, and z, : X z, (73) L + l z Y + h. (74) L + l Assuming the sag of the fist mio can be witten in the fom z b g b g, (75) z (, ) z d, z d, +z b g z b, g whee z and ae given b Eqs. (73) and (74). Then, the sag of the fist mio can be witten as, { } 1 z(, ) [ u X( u) ] du+ [ v Y v ] dv L + l ) (76) o eplicitl afte evaluating the integals in Eq. (76) 1 z(, )( L+ l ) ( + ) + π ef ( ). (77) π 4 ep 3ep ef + ef 3 3 Finall, an epession fo the sag of the second mio follows fom Eq. (7): L NM b g b g l m X X QP 1 b g b g 1 m Y Y Y Y h h Z( X, Y) z(, ) L + O P whee the and tems ae eliminated fom Eq. (78) b solving fo the invese of the magnifications, as descibed in the discussion afte Eq. (6). Some inteesting applications of two-mio beam shaping optics ae pesented in Refs. 9 and 1. These applications include tansfomation of a linea amp beam pofile of one slope to anothe slope with diffeent off-set distances fom the o aes and development of an unobscued two-mio lase pofile shaping sstem. bg (78)

15 4. SUMMARY AND CONCLUSIONS A geometical optics theo fo the design of lase beam shaping sstems has been pesented. This theo of lase beam shaping is based on consevation of adiant eneg within a bundle of as, which is also known as the intensit law fo popagation of a bundle of as, the a tace equations, and the constant optical path length condition fo cases when the contou of the incident wavefont is maintained as the beam passes though the sstem. This theo has been used to develop equations fo the shapes of the optical sufaces fo one-mio, two-plano-aspheic lenses, and two-mio beam shaping sstems, that will tansfom a Gaussian input beam into a unifom iadiance output beam. Paametic equations, which define the contou of the two-mio sufaces equied to tansfom a Gaussian input beam with an elliptical coss-section into a cicula output beam with unifom iadiance, ae pesented. REFERENCES 1. D. L. Sheal, Geometical Methods, in Lase Beam Shaping: Theo and Techniques, Macel Dekke, Inc., New Yok, in pint.. W.T. Silfvast, Lase Fundamentals, Cambidge Univesit Pess, New Yok, J.H. McDemit and T.E. Hoton, Reflective optics fo obtaining pescibed iadiative distibutions fom collimated souces, Applied Optics 13, pp , J.H. McDemit, Cuved eflective sufaces fo obtaining pescibed iadiation distibutions, Ph.D. Dissetation, Univesit of Mississippi, Ofod, P.W. Malak, Two-mio unobscued optical sstem fo eshaping the iadiance distibution of a lase beam, Applied Optics 31, pp , J.L. Keuze, Coheent light optical sstem ielding an output beam of desied intensit distibution at a desied equiphase suface, U. S. Patent 3,476,463, 4 Novembe, P.W. Rhodes and D.L. Sheal, Refactive optical sstems fo iadiance edistibution of collimated adiation: thei design and analsis, Applied Optics 19, pp , P.W. Rhodes, Design and analsis of efactive optical sstems fo iadiance edistibution of collimated adiation, M. S. Thesis, The Univesit of Alabama in Bimingham, D. F. Conwell, Non-pojective tansfomations in optics, Ph.D. Dissetation, Univesit of Miami, Coal Gables, D. F. Conwell, Non-pojective tansfomations in optics, Poc. SPIE 94, pp. 6-7, W. Jiang, D.L. Sheal, and J.C. Matin, Design and testing of a efactive eshaping sstem, Poc. SPIE, pp , W. Jiang, Application of a lase beam pofile eshape to enhance pefomance of hologaphic pojection sstems, Ph.D. Dissetation, The Univesit of Alabama at Bimingham, W. Jiang, D.L. Sheal, and K.M. Bake, Optical design and testing of a hologaphic pojection sstem, Poc. SPIE 15, pp. 44-5, P. Mououlis and J. Macdonald, Geometical Optics and Optical Design, Ofod Univesit Pess, New Yok, M. Bon and E. Wolf, Pinciples of Optics, Fifth Edition, Pegamon Pess, New Yok, S. Solimeno, B. Cosignani, and P. DiPoto, Guiding, Diffaction, and Confinement of Optical Radiation, p. 49, Academic Pess, Olando, A.K. Ghatak and K. Thagaajan, Contempoa Optics, p. 4, Plenum Pess, New Yok, Bon and Wolf, p D.G. Koch, Simplified Iadiance/Illuminance Calculations in Optical Sstems, pesented at the Intenational Smposium on Optical Sstems Design, Belin, Geman, Septembe 14, 199, and published in Poc. SPIE , D.G. Bukhad, and D.L. Sheal, Simplified fomula fo the illuminance in an optical sstem, Applied Optics, pp , D.G. Bukhad and D.L. Sheal, A Diffeent Appoach to Lighting and Imaging: Fomulas fo Flu Densit, Eact Lens and Mio Equations and Caustic Sufaces in Tems of the Diffeential Geomet of Sufaces, Poc. SPIE 69, pp. 48-7, O.N. Stavoudis, The Optics of Ras, Wavefonts, and Caustics, pp. 8-84, Academic Pess, New Yok, Milita Standadization Handbook - Optical Design, section 6.8, volume MIL-HDBK-141, Defense Suppl Agenc, Washington 5, DC, 5 Octobe, Stavoudis, Chapte II. 5. Silfvast, pp CODE V is a egisteed tademak of Optical Reseach Associates, 38 E. Foothill Blvd., Pasadena, CA OSLO is a egisteed tademak of Sinclai Optics, Inc., 678 Palma Road, Faipot, NY ZEMAX is a egisteed tademak of Focus Softwae, Inc., P.O. Bo 188, Tucson, AZ W.J. Smith and Genesee Optics Softwae, Inc., Moden Lens Design, p. 51, McGaw-Hill, New Yok, A. Meddle and R. Mead, A simple method fo function minimization, Compute Jounal 7, pp. 38, N.C. Evans and D.L. Sheal, Design and optimization of an iadiance pofile shaping sstem with a genetic algoithm method, Applied Optics 37, pp , W. Jiang and D.L. Sheal, Development and Testing of a Refactive Lase Beam Shaping Sstem. Poc. SPIE ,.