Historical perspective of laser beam shaping

Transcription

1 Histoical pespective of lase beam shaping David L. Shealy Univesity of Alabama at Bimingham Depatment of Physics, d Avenue South, CH30 Bimingham, AL USA ABSTRACT An oveview of the histoy and cuent pactices of lase beam shaping is pesented. When diffaction effects ae not impotant, geometical methods fo lase beam shaping (ay tacing, consevation of enegy within a bundle of ays, and the constant optical path length condition) can be used to detemine system configuations, including aspheic elements and spheical-suface GRIN lenses, which ae equied to tansfom an input lase beam pofile into a moe useful fom of illumination. This pape also summaizes applications of these techniques to the optical design of a two-planoaspheic lens system fo shaping a otationally symmetic Gaussian beam, a two-mio system with no cental obscuation fo shaping an elliptical Gaussian input beam, and a thee-element spheical suface GRIN system fo shaping a otationally symmetic Gaussian beam. Keywods: lase; beaming shaping; geometical optics; optical design; iadiance mapping. INTRODUCTION Ealy thoughts of beam shaping in non-lase systems can be taced to befoe the days of Achimedes and his buning glass,, whee optics was epoted to concentate -- to incease the powe density of -- sola adiation. The liteatue is 3, 4, 5, 6, 7 ich with epots of vaious optical systems used as sola collectos. Welfod and Winston 8 have pesented a 9, 0 good accounting of non-imaging (non-focusing) optics used as sola collectos, including an ideal light collecto, which concentates a beam by the maximum amount allowed by phase space consideations. Bukhad and Shealy have used a diffeential equation method to design a eflecting suface, which distibutes the iadiance ove a eceive suface in a pescibed manne. McDemit and Hoton, 3 pesented a genealized technique fo designing a otationally symmetic eflective sola collecto, which can heat the collecto suface in a pescibed manne. Beam shaping has also been used in opto-electonics to achieve maximum powe tansfe between a mico-optics light souce and an optical fibe, 4, 5 in adiative heat tansfe, 6, 7, 8 in illumination applications, 9, 0,, 3, 4 and fo eflecto synthesis. Fo illumination applications using a lase beam, such as in hologaphy, mateials pocessing, and lithogaphy, it is vey impotant to unifomly illuminate the taget suface. Both eflective, 5, 6, 7 and efactive 8, 9, 30, 3, 3, optical systems have been used to shape lase beam intensity pofiles. McDemit and Hoton 5 use consevation of enegy within a bundle of ays to design otationally symmetic eflective optical systems fo illuminating a eceive suface in a pescibed manne using a non-unifom input beam pofile. Malyak, 6 Shealy and Chao 7 have designed a two-mio lase pofile shaping system with ectangula symmety and no cental obscuation. Keuze 9 has patented a coheentlight optical system using two aspheical sufaces to yield an output beam of desied intensity distibution and wavefont shape. Rhodes and Shealy 30 deived a set of diffeential equations using intensity mapping and the constant optical path length condition to calculate the shape of two-aspheical sufaces of a lens system that expands and convets a Gaussian lase beam pofile into a collimated, unifom iadiance output beam. Using thei method, two-plano-aspheical lenses have been designed, fabicated and used fo lase beam shaping in a hologaphic pojection system. 33, 34, 35 Hoffnagle and Jeffeson 3 intoduced convex aspheical sufaces fo ease of fabication and a continuous oll-off of the output beam pofile fo moe contol of the fa-field diffaction patten into thei design of a efactive lase beam shaping system. Optical design of lase beam shaping systems has evolved consideably fom the ealy wok of Fieden 8 and Keuze 9 duing the 960 s to the contempoay wok of many summaized in Refs. 36, 37, and 38. This ealy wok aticulated well the goals of some contempoay lase beam shaping applications. Namely, Fieden and Keuze sought to define an optical system that would tansfom an input plane wave with a Gaussian iadiance pofile into an output plane wave with unifom iadiance. Consevation of enegy along a bundle of ays was used to establish a non-linea mapping of the ay coodinates between the input and output planes. Fieden shows that the phase of the beam ove the Othe autho infomation: dls@uab.edu ; Telephone: ; Fax: Copyight 00 Society of Photo-Optical Instumentation Enginees. This pape will be published in Poc. SPIE 4770 and is made available as an electonic pepint with pemission of SPIE. One pint o electonic copy may be made fo pesonal use only. Systematic o multiple epoduction, distibution to multiple locations via electonic o othe means, duplication of any mateial in this pape fo a fee o fo commecial puposes, o modification of the content of the pape ae pohibited.

2 output plane may vay by 0λ afte edistibution of the beam iadiance. Theefoe, fo lase beam shaping applications when the output beam phase is impotant, a second optical element must coect phase distotions intoduced by the iadiance edistibution. Fieden computed the shaping of an aspheical efacting suface that would e-collimate the output beam paallel to the optical axis and also to the input beam. Keuze imposed the constant optical path length condition fo all ays passing though the beam shaping optics to contol phase vaiation of the output beam. Unfotunately, optical design and fabication technologies wee geneally not adequate until the 980 s to pemit ealistic design, analysis, fabication, and testing of lase beam shaping systems. Today, beam shaping is the pocess of edistibuting the iadiance and phase of a beam of optical adiation. The iadiance distibution defines the beam pofile, such as, Gaussian, multimode, annula, ectangula, o cicula. The phase of the output beam detemines its popagation popeties. Contempoay lase beam shaping systems can be gouped into two functional categoies: field mappes and beam integatos. A field mappe tansfoms a known input beam into a desied output beam in a pescibed manne, can be effectively lossless, and woks well fo single-mode beams. A beam integato beaks the input beam into a lage numbe of facets by a lens-aay, and then, ties to spead the enegy within each facet ove the output egion. The output beam pofile is a sum of the diffaction pattens of each lens-aay apetue. Beam integatos wok well fo multimode beams whee the input pofile may be unknown. Optical design of beam shaping systems can be achieved using eithe physical o geometical optics. Thee is no single beam shaping method that can be used fo all applications. Guidance in choosing a beam shaping technique is discussed in Chapte of Ref. 37. Fo single-mode Gaussian beams calculating the paamete β will help detemine the quality of solution available and whethe geometical o physical optics methods should be used β π Y f λ 0 0 = () whee λ is the wavelength, 0 is the beam adius o waist, Y 0 is half-width of the desied output dimension, and f is the focal length of the focusing optic, o the woking distance fom the optical system to the taget plane fo systems without a defined focusing optics. Fo simple output geometies, such as, cicles and ectangles, the following ules of thumb have been developed: 39 If β <4, a beam shaping system will not poduce acceptable esults. When 4 < β < 3, diffaction effects ae significant and should be pat of design of beam shaping systems. When β > 3, geometical methods should be adequate fo design of beam shaping systems. This pape will summaize the histoical development and application of the geometical methods 40, 4 fo designing eflective and efactive field mapping systems. Field mapping is basic to all beam shaping, since one seeks to design a set of optical elements that map an input field into a desied output optical field. A histoical pespective of the theoy of geometical methods fo design of a lase beam shaping system is pesented in section. A bief oveview is pesented of the optical design pocess of incopoating the geometical optics intensity law fo popagation of a bundle of ays and the constant optical path length condition into the ay tace equations, and then, of detemining the geometical contou of seveal sufaces so that the beam shaping design conditions ae satisfied. Optimization-based techniques, such as, genetic algoithms (GA), have also been shown to be effective methods fo design of lase beam shaping systems. 4 Thee applications - a two-plano-aspheic lens system fo shaping a otationally symmetic Gaussian beam, a two-mio system with no cental obscuation fo shaping an elliptical Gaussian input beam, and a thee-element GRIN system with spheical sufaces fo shaping a otationally symmetic Gaussian beam ae discussed. These applications have been selected to illustate how the geometical methods fo optical design of lase beam shaping systems ae applied to a ange of configuations. Section 3 pesents a bief intoduction to the benefits of using beam shaping optics when the oveall efficiency inceases if the detecto is unifomly illuminated.. GEOMETRICAL METHODS In ode to optimize the iadiance within an optical system, the optical field must be detemined thoughout the system. 43, 44 The optical field is a local plane wave solution of Maxwell's equations o the scala wave equation. Fo an isotopic, non-conducting, chage-fee medium, the optical field may be witten as: u( ) = u 0 ( )exp ik 0 S( ), ()

3 whee k 0 =ω/c=π/λ 0 is the wave numbe in fee space; u 0 () and S() ae unknown functions of. Requiing u() fom Eq. () satisfy the scala wave equation leads to the following conditions which must be satisfied: b S g = n (3) u S u + u S = u S = u na = Ia = 0 (4) whee n is the index of efaction, and I is the enegy density of the field times the speed of popagation within medium. Equation (3) is known as the eikonal equation and is a basic equation of geometical optics. The sufaces S(x,y,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 wavefont. Equation (4) expesses consevation of adiant enegy within a bundle of ays and is known as the geometical optics intensity law fo popagation of enegy. Accoding to geometical optics, the phase and amplitude of the optical field ae evaluated independently. Fist, the ay paths ae evaluated thoughout the optical system with ay tacing. Then, the phase of the optical field is computed fom the optical path length of the ays passing though the system. The amplitude (o intensity) of the optical field is computed fom the, 45, 46 density of ays at any point within the system by monitoing the intensity vaiations along each ay. Fo beam shaping systems with collimated input and output beams as illustated in Figue, a useful expession fo the enegy within a bundle of ays 47 as it passes though the system follows by integating Eq. (4) ove efeence planes (o wavefont) nomal to the input and output beam and then applying Gauss theoem I dw = I dw. (5) in Equation (5) expesses consevation of enegy along a bundle of ays between input element of aea and the coesponding output element of aea on the wavefont o the efeence planes nomal to the beam. Equation (5) says that the intensity times the coss-sectional aea of the beam is constant along the beam as it popagates though the optical system and is a basic equation used fo the optical design of lase beam shaping systems. Fo some beam shaping configuations, it is necessay to intoduce the consevation of enegy condition into the optical design by using efeence sufaces, such as fo detectos, which ae cuved and/o have an abitay oientation with espect to the diection of the beam popagation. In these cases it is necessay to take into account pojecting the element of aea of a efeence suface pependicula to diection of beam popagation when applying Eq. (5). The total enegy of the beam must also be conseved as the beam popagates though the system. out Figue. Schematic layout of a lase beam pofile shaping system. Ray tacing 48 is widely used to simulate the pefomance of both imaging and non-imaging optical systems. By assigning each incoming ay equal enegy density, then by counting the numbe of ays cossing a unit of aea within the optical system, the iadiance can be computed thoughout the optical system. Kock 45 epots a method to simplify photo-adiometic calculations of optical systems by using a efeence sphee and ay tacing. The flux flow equation, 46, 49, 50 offes an altenate appoach fo evaluating the iadiance within an optical system. The flux flow equation along with the ay tace equations ae used to monito the change in size of an element of aea of a bundle of ays 47 as the wavefont popagates though the optical system. The flux flow equation depends on the beam paametes and the shape/oientation of the optical sufaces and allows the iadiance to be computed along a ay path as it popagates

4 though an optical system. The flux flow equation can also be consideed as a diffeential equation of the optical suface contou, which can be solved if the input and output beam pofiles ae known. In ode to accomplish the oveall beam expansion and pofile shaping as illustated in Figue, two optical elements ae equied. These optical elements may be eithe lenses o mios. The fist optical element typically expands and shapes the beam pofile to satisfy consevation of enegy accoding to Eq. (5), and the second optical element e-collimates the ays so that the constant optical path length condition is satisfied by the system, as illustated by the two-lens beam shaping system shown in Figue. Optical design of a lase beam shaping system seeks to define the optical components adequately so that the system can be analyzed, fabicated, and tested. This geneally equies specification of the shape and spacing between the optical sufaces as well as the index of efaction of all the media... Rotationally Symmetic Systems When the incident beam entes the system at a distance fom the optical axis with an intensity distibution I in (), this beam leaves the optical system at a adial distance R fom the optical axis with a powe density of I out (R). Integating Eq. (5) ove the input and output planes gives π max Rmax dθ I () d= dθ I ( R) RdR. in When I out ( R ) is constant, the ight hand side of Eq. (6) integates to give the total powe, πi out R max, passing though the system. One of the two paametes, R out o I out, may be abitaily specified. The othe is detemined fom Eq. (6) to give max max max = in out = in Iout R 0 max 0 π, o R I d I I d. Once I out is known, the adius of an abitay ay in the output beam is given by: out (7) =± I R in I u udu (8) out 0 whee is the adius o distance fom the optical axis of the ay in the entance pupil. The positive sign is used in Eq. (8) when the ays ae divegent fom the fist lens, i.e., the lens suface s is concaved, as shown in Figue. The negative sign is used when the ays ae convegent fom the fist lens, i.e., the lens suface s is concaved, and the ays come to a focus and coss the optical axis befoe eaching the second lens. In the fundamental, Gaussian TEM 00 mode of a lase when the cental intensity has been nomalized to unity, the input beam pofile is given by I = exp in (9) whee 0 is the beam adius o waist. Altenatively, when P is the total powe of the incident beam, the input intensity pofile is given by Pexp ( ) 0 I in = (0) ( π 0 ) { exp ( max 0) } whee max is the adius of input apetue o lens. It is also helpful to note some authos chaacteize the input Gaussian beam pofile in tems of anothe constant (α) equal to half the beam waist, i.e., α = 0 /. When the input beam pofile is given by Eq.(9), the adius of beam in the exit apetue R can be evaluated fom Eq. (8) to obtain exp( 0 ) =± R max exp( ) max 0 R (6). ()

5 Equation () is used duing the optical design pocess to educe the numbe of independent vaiables when solving fo the shape of the eflecting o efacting sufaces of the beam shaping system. The consevation of enegy condition and the constant optical path length condition can be solved simultaneously with the ay tace equations fo R(), z(), Z(R) when design paametes, such as, n, d, t, t fo the two-element system shown in Figue, ae given. It is inteesting to note that seveal authos 6, 7, 40 have shown that the sag of two optical elements of a lase beam shaping system can be expessed as a function of z( ) = f ( ) d+ C () Z ( ) = z( ) + g() (3) whee C is a constant, and f() and g() ae functions defining the optical configuations.... One-mio beam shaping system Figue. Geometical configuation of a two-lens lase expande. (Fom Ref. 33.) Conside the otationally symmetic geomety of a one-mio beam shaping system shown in Figue 3. The adiation is incident upon the mio suface s defined by z = z (). The equation of the eceive (detecto) suface S is given by Z = Z (R). The iadiance of the incident beam, I in (), is incident upon a cicula ing about the z-axis of aea da = πd and is eflected to a cicula ing on the eceive suface S of aea da = πr [ + (dz/dr) ] /. The input adiation is collimated (paallel to optical axis) with a known intensity pofile. It is desied to iadiate the eceiving suface (detecto) with a pescibed intensity distibution without placing any conditions on the shape of the wavefont of the output beam. The ay tace equation connecting the mio suface s with the eceiving suface S in the -z plane is given by which can be ewitten as R A z = = ZR z A z b g b g b g Applying the diffeential enegy balance Eq. (5) to this poblem gives z b g b g. (4) R z + Z z z + R = 0. (5) π = π + = π + I d I R R in out dr dz I R R dz dr dr out. (6) whee the tem (dr/d) can be displayed as will be needed in Eq. (7) below.

6 Figue 3. Geometical configuation of a one-mio beam pofile shaping system. Recall that I in (), I out (R), and Z(R) ae known functions of thei espective vaiables, and z () is an unknown function at this point of the analysis. Also, note that the ay tace Eq. (4) expesses a mapping between sufaces s and S, which implies that R is a function of. By implicit diffeentiation of Eq. (5) with espect to, the diffeential enegy balance equation (6) can be incopoated into the ay tace equation to give: z z dr = 0 d. (7) ( R ) ( z ) z zz ( z ) whee the chain ule fo diffeentiation of the function of a function tem (dz/d) and Eq. (5) have been used. The tem (dr/d) can be eliminated between Eqs. (6) and (7) to give the following diffeential equation which detemines the sag z () of mio suface s: z z dz + z I + ( + ) in z z dr =. (8) z ( R ) Iout ( R) R dz + dr Equation (8) is equivalent to Eq. 3.4 of Ref. o Eq. 3 of Ref. 3. When appopiate bounday conditions ae given, Eq. (8) can be solved fo the shape of the mio used to illuminate the eceive suface S with a pescibed intensity I out (R) fo a given souce intensity pofile I in (). Refeences and 3 develop an extension of this analysis to two-mio intensity pofile shaping systems. A numbe of specific solutions fo both one- and two-mio systems ae given in Refs. and 3 including two lase beam pofile shaping systems, such as, unifom illumination of a plane pependicula to the incident beam using a one-mio system fo an input Gaussian beam, Fig. 7.3 of Ref., and with a two-mio system, Fig. 7.9 of Ref..... Optical design of two-lens beam shaping system A summay of the histoical development of the optical design of a two-lens lase beam shaping system with otational symmety is summaized in this section. By shaping two optical sufaces, it is possible to expand and shape the beam pofile to satisfy the geometical optics intensity law and also e-collimate the ays passing though the system so that the constant optical path length condition is satisfied. Optical design method used by Fieden Fieden 8 descibes how to design an optical system fo conveting the plane wave fom the fundamental Gaussian mode of a lase to a plane wave with unifom iadiance ove a equied coss section with all the powe of the incident beam. Fieden assumes that the iadiance of the input beam is given by Eq. (0), expessed in tems of 0 = α. Then, in a plane B a known distance fom input plane A, the iadiance is unifom. Using geometical optics, it follows that light

7 within any ing of adii, + on plane A will stike plane B within anothe ing of adii R, R+ R whee the path fo each ay height R () follows fom the geometical optics law of intensity o Eq. (8) which is in this case R. (9) The iadiance is unifom ove the plane B, but the adiation is not a plane wave at B, since R () is nonlinea function of. Theefoe, a second optical element is equied to achieve a plane wave ove the plane B. An expanded view of the two plano-aspheic lenses used fo beam shaping is shown in Figue 4. Fieden notes R ( ) R( ) tanθ = (0) Z R z t z exp( α ) =± R max exp( α ) max whee the appoximation [ Z(R) t ] has been made in witing the ight-hand side of Eq. (0). The slope of the suface s is given by dz ( ) tanθ = z. () d Figue 4. Expanded view of geometical configuation of lase beam shaping system. Applying Snell s law to ays efacted at z () expessed in tems of tanθ gives tanθ = nz n z. () Using tigonometic identity fo tan(θ - θ ) = tanθ with Eqs. () and () allows Eq. (0) to be witten as a nonlinea, fist-ode diffeential equation fo z (), the sag of suface s nz z n z R () R = tanθ =. (3) nz + n z Z( R) z( ) t z Equation (3) can be solved numeically using Eq. (9) fo the shape of the aspheic lens suface s, which will unifomly illuminate the plane B. Fieden pesents an appoximate solution of Eq. (3) by assuming t >> z such that (t z) t and (n )z. Then, Eq. (3) can be witten as ( n ) z R( ) =, (4) + nz t which can be witten as a quadatic equation in z

8 Equation (5) has the following solutions ( ) n R z t n z + R = 0 (5) ( ) ± ( ) 4 ( ) n( R ) t n n t n R z = Assuming (n - ) t > n (R - ), the squae oot in Eq. (6) can be expanded to yield n( R ) t( n ) ± t( n ) +... n t R z = n R t n. (6) whee the negative sign is Eq. (7) was used to give a positive slope of the suface s. Integating Eq. (7) with Eq. (9) fo R () gives the following expession fo z (): R max z = exp( u α ) du + t. (8) ( ) t n exp ( 0 α ) 0 t n Fieden uses Simpson s ule to compute numeical values fo the sag of suface s fo a beam shaping optics with the following paametes: 0 =R max = mm; t=0mm; t = t = 5mm, α =.4mm, and n =.5. Fieden notes that thee is maximum optical path length vaiations ove the output pupil plane of this system of 0λ. Theefoe, fo most lase applications, a second beam shaping optic must be used to e-collimate ays ove the output pupil. Fieden seeks to detemine a functional elationship fo the sag Z (R) of suface S such that any ay passing though the system leave paallel to the optical axis. Applying Snell s law at suface S gives sin n sin nz Z whee this fom of Snell s law assumes the output beam is paallel to the optical axis. Noting that Θ = θ + Θ pemits witing the following expession sin Θ = sin θ +Θ = sinθ cosθ + cosθ sin Θ (7) Θ = Θ = +. (9) nz z + n z nz n z + + Z = + + z + Z + z + Z Combining Eqs. (3), (9), and (30) lead to the followin g esults { } ( nz z n z Z nz n z n Z z. (30) = + ). (3) which is consistent with Eq. () of Ref.30. It is inteesting to note that when n = n, then Z = z is a solution to Eq. (3) 30, 3 as epoted in the liteatue. Fieden pesents an quadatic equation fo Z and a numeical algoithm fo computing Z(R) so that optical path diffeence ove the entie pupil is now less than λ/0. It is inteesting to note that caeful analysis of Figue 4 leads to the conclusion that when both lenses have the same index of efaction and when the input and output ays ae paallel to the optical axis, then the slope of the two aspheical sufaces must be equal along a given ay. Theefoe, futhe analysis of Eq. (3) does not seem to elate to pactical lase beam shaping systems. Optical design method used by Kueze Keuze 9 also developed a geometical optics-based method to edistibute the ays of an input lase beam to yield an output beam with a pescibed intensity distibution while maintaining constant optical path lengths between the input and output wavefonts. Keuze has pesented design equations fo a two-element efactive system that will tansfom a collimated input Gaussian beam into a collimated output beam with unifom iadiance as illustated in Figue. Equation (9) gives the intensity of an input beam in Gaussian TEM 00 mode with cental intensity nomalized to unity. Rays enteing the system at a adial distance leave the system with an output intensity of I out (R) at a adial distance R

9 as given by Eq. () whee the positive solution of R is used fo the configuation consideed in Figue. Based on the geometical configuation shown in Figue 4, R = R (, s S)sinθ. (3) whee R= R + ( Z z). Eliminating R between Eqs. () and (3) gives ( 0) e + R (, s S)sinθ Rmax ( max 0 ) e = 0 (33) which must be satisfied by values of on the efacting suface s in ode to edistibute the input beam pofile accoding to the equiement of consevation of enegy. When both the input and output wavefonts ae plana, all ays passing though the beam shaping system must have the same optical path length. Equating the optical path length of an axial and non-axial ay leads to d( n ) + R ( ncosθ ) = 0. (34) Snell s law fo ays efacted at suface s towads S is given by sinθ = sinθ = sin θ + θ = sinθ cosθ + sinθ cosθ. (35) n Fom Figue 4, tanθ is the slope of sufaces s and S, o tan dz dz R d dr (36) A goal of this analysis is to expess z () and Z (R) in tems of n, d, and. Keuze poceeds by eliminating R between Eqs. (3) and (34) to give ( n cosθ ) ( n ) d =. sinθ ( R ) (37) Dividing Eq. (35) by cosθ and displaying tanθ gives sinθ tanθ =. ( cosθ n) (38) Eliminating θ between Eqs. (37) and (38) will give a fist ode diffeential equation fo z (). Keuze obtains this esult by squaing Eq. (37) and adding to (n ) ( n ) d ( R ) ncosθ + n cos θ cos θ ncosθ + n + n = + n = sin θ Noting that the ight-hand side of Eq. (39) is the squae of the invese of the ight-hand side of Eq. (38) allows one to wite the following expession fo the slope of sufaces s and S tanθ =. (40) ( n ) d ( n ) + ( R ) Integating the fist Eq. (36) with Eq. (40) leads to an expession fo z () which defines s d z ( ) = (4) 0 ( n ) d ( n ) + ( R ) whee Eq. () expesses R (). Similaly, integating the second Eq. (36) with Eq. (40) leads to an expession fo Z(R) which defines S sin θ.(39)

10 Z ( R) = R dr (4) 0 ( n ) d + ( R ) ( n ) whee inveting Eq. () expesses max R 0 0 ln e R = Rmax In summay, the fist lens is a plano-aspheic element whee the sag of the aspheic suface is defined by Eq. (4) using Eq. () fo R (). The second lens is an aspheic-plano element whee the sag of the aspheic is defined by Eq. (4) whee (R) is given by Eq. (43). Keuze notes that if d is sufficiently lage, then the tem (n ) in Eqs. (4) and (4) can be neglected. Then, one has z ( ) R d (43) ( n ) d. (44) Keuze also pesents an example solution fo the shape of the aspheical sufaces by tabulating z () and Z (R) fo n =.57, d = 50mm, max = R max = 5mm, and 0 = 4mm. In this case, ninety-pecent of the input beam is incident upon the beam shaping optics. Keuze futhe notes that the fist lens can be a focusing, convex-aspheic element whose sag can be evaluated by eplacing R by -R in the above equations. Optical design method used by Rhodes, Jiang, and Shealy Both the optical design methods of Fieden and Keuze involve solving couple diffeential equations o two integal equations fo the sag of the two efacting sufaces. Rhodes, Jiang, and Shealy 30, 33 simplify the optical design pocess by showing that the sag of two aspheic sufaces satisfy equations in the fom of Eqs. () and (3). Fo moe details and application of these esults, see Refs. 3, 34, 35, 40, and 4. Conside the configuation of a efacting lase beam shaping system shown in Figue. The two cuved sufaces ae used to satisfy the lase beam shaping design conditions. Rays ae efacted at suface s accoding to Snell's law. The ay tace equation of efacted ay A taveling fom the point (, z) on suface s to the point (R, Z) on suface S is given by + ( ) R - A z n z n = =. (45) ( Z z) A z nz + + z n Equation (45) can be expessed as a quadatic equation in z and solved to yield z = 0 ( R )( Z z) ± n ( Z z) + ( R ) n ( Z z) n ( R ). (46) The positive solution fo z is used fo the lens configuation shown in Figue whee the fist lens is divegent. Fo this system, the height of the ay R at the second lens with entance pupil height is computed Eq. () with the positive solution. The tem (Z - z) in Eq. (46) is detemined by the constant optical path length condition ealized by setting the axial optical path length equal to that of a geneal ay ( R ) + ( Z z) = n( Z z) d( n ), (47) which is a quadatic equation fo the tem (Z z) as a function of the entance pupil apetue adius. Afte squaing Eq. (47) and collecting tems, the solution of the esulting quadatic equation is

11 nn ( ) d+ ( n ) d + ( n )( R ) ( Z z) = (48) n 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 = 0. It is inteesting to note that Eq. (46) pemits z to be expessed as a function of, thus, enabling z () to be evaluated by integation, as illustated in Eq. (). Refeences 33 and 34 pesent esults fo design, fabication, and testing of a two-lens lase beam shaping system simila to configuation shown in Figue...3. Analysis, fabication, and testing of two lens beam shaping systems Seveal optical design methods fo calculating the sag of the two aspheical lens sufaces equied to shape an input Gaussian beam have been discussed in the pevious section. The design methods of Rhodes, Jiang, and Shealy have been used to develop a pototype lase beam shaping system, which has been designed, fabicated and tested. 33 The lens thickness is not involved in the design diffeential equations, but is an impotant facto to be consideed, since it elates to the enegy absoption by the optics. The lens thickness of each element has been assumed to be 0mm fo this system. Anothe system paamete, the distance between the two lenses is impotant fo optical design, fabication, and testing of the system. The lage the spacing between the two elements, the smalle the suface cuvatue of each element equied to satisfy the design conditions. Howeve, if the element spacing is too lage, the system will be difficult to assemble and test. The distance between these elements has been abitaily chosen fo this system to be five times the diamete of the lens elements. Table. Suface paametes of a HeCd (44.57nm) lase pofile shaping system whee the distance between the pimay and seconday lens is 50mm. (Fom Ref. 40) Lens Suface Paametes Pimay Mio SecondayMio Diamete (mm) Vetex Radius (mm) Index of Refaction.4396 (CaF ).4396 (CaF ) Thickness (mm) Conic Constant, κ A 4 (mm -3 ) x x 0-6 A 6 (mm -5 ) x x0-9 A 8 (mm -7 ) x x0-0 A 0 (mm -9 ) x x0 - A (mm - ) x x0-5 The index of efaction of the lenses affects the shape of efacting sufaces s and S. Fo this application the lens elements must have a high tansmission fo the wavelength of the lase being used [44.57mm fo Helium Cadmium (HeCd) lase] with this pototype lase beam shaping system. Thee ae seveal mateials with good tansmission popeties at 44.57nm 5 Acylic (PMMA) Plastic, Calcium Fluoide (CaF ), Cown Glass (BK7), Lithium Fluoide (LiF), Fused Quatz, and Fused Silica. Acylic Plastic is difficult to polish. Fused Quatz, Fused Silica, and Cown Glass ae difficult to machine with a single-point diamond lathe as equied in making these aspheic sufaces. Lithium Fluoide contains defects and is difficult to wok with. Calcium Fluoide has good tansmission popeties (~95%) within the ange of nm, can be machined with a single-point diamond lathe, and is also the least expensive of this goup of mateials. Theefoe, CaF has been selected as the lens mateial fo this lase beam pofile shaping system. The index of efaction of CaF is a function 5 of the wavelength of the light being used: n = + 3 i= A λ i λ λ (49) whee λ is measued in units of micons (µ) and λ = µm; λ = µm; λ 3 = µm; A = ; A = ; A = Using Eq. (49), the index of efaction of CaF has been calculated to be.4396 at the HeCd lase wavelength of 44.57nm. i

12 Fo this lase pofile shaping system, the lase beam is expanded fom 6mm to 5mm in diamete while the beam intensity pofile is flattened. Afte solving the diffeential equations, a non-linea least squaes fitting pocess was used to fit the lens suface data to the conventional optics suface equation N c i z = + A i (50) + + c i κ = whee C (vetex cuvatue), κ (conic constant), and A i (coefficients of the polynomial defomation tems) ae suface paametes that ae detemined by the fitting pocess fo each suface. The data fo sufaces s and S obtained fom solving the diffeential equations has been fit to many diffeent expessions fo the optics suface equation. Moe aspheic tems will geneally give a bette fit with smalle least squaed eos. Howeve, it is easie to fabicate and test sufaces with a non-zeo conic tem and a small numbe of polynomial defomation tems. The data in Table with non-zeo conic constant plus five defomation tems epesents a compomise between fitting accuacy, optical modeling pefomance, and ease of fabication...4. Analysis of Lens Pefomance The optical pefomance of the optical system defined by Table has been modeled and compaed to the design conditions of the lase pofile shaping system. Applying the flux flow equation 46 to this lens system, the intensity of a coss-section of the output beam has been evaluated. A caeful computation of the aea unde the sufaces geneated by otating aound the optical axis the input and output intensity pofiles given in Figue 7 of Ref. 40 shows that the total powe is conseved fo this lase beam pofile shaping system. The optical path diffeence (OPD) of the output beam fo the shaping system defined in Table has also been evaluated ove the apetue. The maximum OPD fo this system is 0.007λ, which coesponds to the absolute OPD of 0.75nm fo the HeCd lase. This demonstates that the shape of the output wavefont has the same shape as the input wavefont, as equied by the constant OPL condition It is also inteesting to conside how this lase beam shaping system would pefom using a diffeent lase wavelength, λ. Since the index of efaction is a function of wavelength as given by Eq. (49) and since these optics have been designed fo a specific n(λ), it is impotant to detemine whethe a set of lase beam pofile shaping optics can be used fo multiple λ. Fo example, if the optics defined in Table wee used with a Helium Neon (HeNe) lase with wavelength 63.8nm, the index of efaction of the lenses would be educed to Then, accoding to Snell's law, the light leaving the pimay lens would not be efacted as stongly as the HeCd light. The sepaation between the two lenses would need to be inceased to accommodate this decease in index of these lenses to insue that the light is incident upon the seconday at the appopiate height so that it will be efacted paallel to optical axis as illustated in Figue 5. Figue 5. Illustates the elationship λ and d. Now, we will deive an expession fo the lens sepaation fo a paticula ay so that this ay will be incident upon suface S at a suitable point such that afte efaction, this ay will leave the seconday lens at the appopiate height R to insue consevation of enegy and to be paallel to the optical axis. The new distance d can be calculated fom Eq. (46), since, R, and z ae known. Equation (46) can be witten as a quadatic equation of (Z z) with the physical solution Z z = b gl NM c h cn hz R + n + n z O QP. (5)

13 Once (Z z) is known, then the coesponding value of d can be detemined by solving Eq. (48) as a quadatic equation fo d: n n d n n n Z z d n c hb g b gc hb g + c hbz zg cn hbr g = 0 (5) with the physical solution b g b g nz z Z z + R d =. (53) n If the lenses ae sepaated by the vetex spacing d of Eq. (53), then the ay leaves the seconday lens paallel to the optical axis. Howeve, evey ay in the entance pupil will equie a diffeent lens spacing d. Calculations have shown that the optical system defined in Table will need a lens spacing of appoximately d = 5.8mm when used to shape the 63.8nm HeNe lase beam pofile. The maximum diffeence between the lens spacing fo all ays is less than 0 micons, which is smalle than alignment eos. Theefoe, it seems easonable to use this lase beam pofile shaping system at multiple wavelengths. The two plano-aspheical lenses of Table has fabicated by Janos Technology, Inc. of Townshend, Vemont, using a single point diamond lathe whee CaF was used as the lens mateial. A scanning video system was used to measue the input and output beam pofiles. A full discussion of these expeimental esults is pesented in Refs. 33 and 34. Thee was appoximately a 0% vaiation of the output beam intensity fo the system fabicated and made duing the ealy 990 s. The output wavefont shape of this two plano-aspheical lens lase beam pofile shaping system has also been evaluated. Imposing the constant optical path length condition equies the output wavefont to have the same geometical shaping as the input wavefont, which is assumed to be plana. The collimated adiation of the input lase beam has a vey small divegence, which means that the beam diamete does not incease o decease with popagation within limits of geometical optics. Fo popagation of the output wavefont of this lase pofile shaping system aound the laboatoy, say 0m, the output beam diamete emained constant. To quantify this esult, Ronchigams 53 of the input and output beams wee ecoded using the video system descibed in this section fo ecoding the beam intensity pofile. 54 Analysis of these Ronchigams indicates that the output beam wavefont has the same shape as the input wavefont. In addition, a HeNe lase was used to illuminate these pofile shaping optics, and the pedictions of section..4 have also been confimed. When using a HeNe lase with 63.8nm adiation, the lens spacing was inceased to 5.mm accoding to the pedictions of Eq. (53)... Non-Rotationally Symmetic Systems Fo a pojective tansfomation 55 in optics, a point in image space can be expessed as a linea function of the coodinates of the object point. Pefect imaging systems, such as, Maxwell's fish-eye lens o stigmatic imaging of sufaces, ae examples of pojective tansfomations in optics. In pactice, abeations ae pesent in many optical 56, 57 systems, and point-to-point imaging is not possible, except to the fist-ode o paaxial appoximation. Conwell notes that all eal optical systems pefom non-pojective tansfomations to some extent. That is, thee is a non-linea dependence between input (o object) and output (o image) coodinates. The edistibution of ays leading to the beam shaping allows fo tansfomation of a Gaussian input beam into a moe unifom (top-hat, Femi-Diac, o supe-gaussian) iadiance output beam equies a non-linea elationship between the input and output apetue coodinates. Theefoe, the geometical methods 40 fo designing a lase beam shaping system ae an example of a non-pojective tansfomation in optics. Conwell notes that the fist element of a lase beam pofile shaping (non-pojective tansfom) system ceates sufficient abeations in the wavefont to estuctue the intensity of the beam afte popagation of the wavefont ove a specified distance. Then, the second element of a lase beam pofile shaping system has suitable contou to estoe the oiginal wavefont shape of the beam. If the pupose of a lase beam pofile shaping system is to unifomly illuminate a suface, then the second element is not needed. Symbolically, a lase beam pofile shaping system may be consideed to be a black box which tansfoms an input lase beam (plane wave) with a Gaussian intensity distibution into an output beam (plane wave) with unifom intensity distibution. The input and output beams have adii and R, espectively, as shown in Figue. Refeence 56 pesents extensive discussion of many types of lase beam shaping systems and daws some inteesting geneal conclusions. In paticula, Conwell povides a seven-step ecipe fo designing two-element systems, which pefom nonpojective tansfomations, such as, lase pofile shaping systems. Since the contents of Ref. 56 ae not widely available in the optics liteatue to the knowledge of this autho, these seven steps ae summaized below:

14 . Wite out diffeential powe expessions fo the intensity distibutions ove the input and output planes. Rectangula Coodinates I ( x, y) dxdy = I ( X, Y ) dxdy (54) in out Pola Coodinates I () d= I ( RRdR ) (55) in out. Use the consevation of enegy to elate the input and output beam paametes. Rectangula Coodinates I ( x, y) dxdy = I ( X, Y ) dxdy in out (56) Input Apetue Output Apetue Pola Coodinates Iin() d= Iout ( RRdR ) (57) Input Apetue Output Apetue 3. Detemine the magnification elating the input and output ay heights. Rectangula Coodinates Assume the intensity functions ae sepaable I ( xy, ) = a xa y (58) in x y (, ) I XY = A X A Y (59) out X Y Allowing fo non-unifom shaping of a lase beam pofile in two othogonal diections, X = m x (x)x and Y = m y (y)y, the ectangula magnifications follow fom combining Eqs. (54), (58) and (59) x ax ( u) du mx ( x) = C C x + A 0 X ( umx( u) ) (60) y ay ( v) dv my ( y) = C3 y C + 0 AY ( vmy( v) ) (6) whee C i ae constants detemined by bounday conditions, such as, the magnification fo a im ay. Pola Coodinates R = m () (6) Iin () d m () = C + (63) I ( ) 0 out m whee C is a constant detemined fom the bounday conditions. 4. Expess the optical path length (OPL) between input and output efeence sufaces of an abitay ay in tems of the OPL of a efeence ay. 5. Detemine the sag z () of the fist element. 6. Detemine the invese magnification elating the ay coodinates at the fist and second elements. 7. Detemine the sag Z (R) of the second element.... Optical design of a two-mio lase beam shaping system The seven-step ecipe of the pevious section has been used to design of a two-mio lase beam shaping system with ectangula symmety and no cental obscuation. Fo moe details, see Refs. 7 and 4. Figue 6 shows the geometical configuation of the two-mio lase beam shaping system consideed. The input and output beams ae collimated and paallel to the optical axis. Assume the input beam iadiance is given by

15 x y Iin( x, y) = exp exp (64) x0 y 0 whee (x 0, y 0 ) ae the beam waist in the x, y diections, and the cental intensity is nomalized to unity. The output beam iadiance is unifom. In this case, explicit expessions fo X(x), Y(y), Z(x,y), and Z(X,Y) have been evaluated. 4 Using these esults, the pefomance of a two-mio lase beam shaping system which tansfoms an input beam with elliptical coss section of 3: atio of beam waist in pependicula diections has been analyzed. 7 The aspheical deviation of mio sufaces fom best-fit sphee has been shown to be 0µm fo a 6mm diamete mio. The optical analysis softwae ZEMAX 58 has been used fo pefomance modeling and toleancing analysis of this system. These esults show that the fist mio suface has a stong aspheical component along the diection of smalle input beam waist and that the output beam pofile emains faily unifom when the mio decentation is less than.5% of the maximum mio suface dimension and tiltation is less than.5 degees about the coodinate axis. Figue 6. Geometical configuation of a two-mio lase beam shaping system with ectangula symmety. (Fom Ref. 7).3. Genetic Algoithm Optimization Method Fo moe complex configuations, it is often difficult to obtain analytical solutions to the lase beam shaping design conditions. Genetic algoithm (GA) optimization methods have been shown to wok well designing lase beam shaping systems, when the meit function contains both discete and continuous paametes. 4 As an example of a moe complex poblem one difficult to solve using moe conventional methods the GA technique has been used to design a gadient index (GRIN) lase beam shaping system. Wang and Shealy 59 solved this poblem using a diffeential-equation design method. Though Wang was able to poduce seveal pefectly good solutions, the GRIN pofiles of the lenses epoted wee detemined solely by solving the diffeential equations, and no constaints wee imposed that the GRIN pofiles of the lenses coespond to those that can be found in common glass catalogs. It follows that a challenging poblem fo the GA optimization method would be to ceate a lase shaping system with spheical-suface GRIN elements, as Wang did, but to do so with the added constaints that catalog GRIN glass types ae used and the numbe of lens element equied is an optimization vaiable. Solution to this poblem will epesent a way to constuct a lase beam shaping system without use of aspheical optics o esoteic GRIN pofiles. 59 The optical design fo this system involves using genetic algoithms (GA) to maximize a lase beam shaping meit function, Eq. (65), within a 6-dimensional paamete space. 60 To solve this optical design poblem, the GA not only must optimize suface shapes of the GRIN elements and thei spacing, but also must detemine the actual numbe of GRIN elements in the solution, up to a cetain limit (fou, in this case), and the type of GRIN mateial fo each element must be selected fom a vendo GRIN glass catalog. This type of poblem distinguishes the GA method fom deteministic methods (i.e., those that ely on deivatives and a smooth, continuous

16 meit function) since the meit function equied fo this poblem depends on a complicated mix of discete and continuous paametes. Fo moe details and applications of these esults, see Refs. 4, 60, and 6. The optical design of this system solves fo the attibutes (adii, thickness, and spacing) of the lens elements, the GRIN glass type fom a catalog, and the numbe of elements needed fo a system of this configuation to satisfy the beam shaping design conditions consevation of enegy within beam and constant optical path length condition. Twenty-six paametes of the lens system ae detemined by the GA optimization. The meit function M includes tems that favo a specific beam diamete, a unifom iadiance output beam pofile, and a collimated output beam. The following meit function satisfies these design objectives and is maximized duing design: M M M Diamete Collimation = = M Unifomity exp exp cos N Q s( RTaget RN ) γ i i= N N Iout ( Ri) Iout R N i= N k= k (65) whee R Taget is the desied adius of the output beam, R N is the adial height of the maginal ay on the output suface, γ i is the angle the i th ay makes with the optical axis, s and Q ae convegence constants used to adjust impotance of diffeent components of meit function duing optimization, and I out (R k ) is the iadiance of ay in output beam. (See Ref. 6 fo a detailed discussion of this GA optimization pocess and constuction of a suitable meit function to use when designing lase beam shaping systems.) The exponential function is used in the meit function, since it peaks stongly as paametes appoach thei design tagets. Also, as the output beam pofile becomes moe unifom, the denominato of Eq. (65) appoaches zeo, and M inceases substantially. In summay, the meit function ewads those systems, which tend to incease the value of M and penalize systems with smalle values of M as the GA optimization seaches thoughout both the discete and continuous paamete space. Afte,367 geneations (iteations), the GA conveges to a thee-element GRIN lens system with all spheical sufaces which is illustated in Figue 7. Fo moe discussion of convegence of the GA optimization method fo application to lase beam shaping systems, see Ref. 60. Scale:.00 Element Element Element 3 Figue 7. Raytace fo the fee-fom GA-designed GRIN shape system. (Fom Ref. 60) 3. APPLICATIONS Lase beam pofile shaping optics ae well suited fo applications whose oveall efficiency inceases when the iadiance 6, 63, 64 ove the detecto (o substate) is unifom, such as in compact hologaphic pojecto systems. These compact

17 hologaphic pojection systems have been epoted to offe a pactical way to make a highly coected mesh o gid patten ove cuved sufaces whee the patten can ange in size fom sub-micon to multi-micon. The lase pofile shaping optics within a hologaphic pojection system enables unifom featues to be witten ove substates of seveal centimetes in diamete. 35 To undestand this incease in system efficiency when using lase beam shaping optics, note that fo a Gaussian beam with iadiance given by Eq. (9), the intensity of the beam deceases to /e 3.5% of its axial value at the beam adius. The effect of this vaiation in beam intensity ove a Gaussian beam is illustated in Figue 8 whee (A) shows significant vaiation in patten densities at the cente and edge of beam fo the same substate (film) and exposue time when a lase pofile shaping optics is not pat of the system, and (B) shows almost unifom patten densities at the cente and edge of beam when lase pofile shaping optics ae pat of the system. Theefoe, when beam shaping optics ae intoduced into a hologaphic pojection pocessing system 35 as illustated, the detecto substate will be unifomly illuminated, and photochemical eactions take place at the same ate ove the entie substate aea, thus, enabling the full beam diamete to be available fo mateial pocessing. Intoducing lase shaping optics into hologaphic pojection pocessing systems have lead to a significant incease in quality of mico-optics fabicated ove the substate. Figue 8. Intefeence pattens poduced by a 4-beam hologaphic pojection pocessing system when illuminated with a Gaussian beam. The image on the left side of the figue was taken nea the cente of the beam, and the image on the ight side of the figue was taken nea the edge (waist) of the beam. The images (A) wee taken when the lase beam pofile optics was not pat of pojection system, and the images (B) wee taken when lase beam pofile shaping optics was pat of pojection system. (Fom Ref. 34)