Ray mapping approach in double freeform surface design for collimated beam shaping beyond the paraxial approximation ChristophBo¨sela,*,NormanG.Workua,HerbertGrossa,b aFriedrich-Schiller-Universita¨tJena,InstituteofAppliedPhysics,Albert-Einstein-Str. 15,07745Jena,Germany bFraunhofer-Institutfu¨rAngewandteOptikundFeinmechanik,Albert-Einstein-Str. 7,07745Jena,Germany Abstract. Numerousapplicationsrequirethesimultaneousredistributionoftheirradianceandphaseofalaserbeam. The beam shape is thereby determined by the respective application. An elegant way to control the irradiance and phase at the same time is from double freeform surfaces. In this work the numerical design of continuous double freeform surfaces from ray mapping methods for collimated beam shaping with arbitrary irradiances is considered. Thesemethodsconsistofthecalculationofaproperraymappingbetweenthesourceandthetargetirradianceandthe 7 subsequentconstructionofthefreeformsurfaces. Bycombiningthelawofrefraction,theconstantopticalpathlength 1 0 andthesurfacecontinuitycondition,apartialdifferentialequation(PDE)fortheraymappingisderived. Itisshown 2 thatthePDEcanbefulfilledinasmall-angleapproximationbyamappingderivedfromoptimalmasstransportwith aquadraticcostfunction. Toovercometherestrictiontotheparaxialregimeweusethismappingasaninitialiterate n a forthesimultaneoussolutionoftheJacobianequationandtheraymappingPDEbyanoptimization. Thepresented J approachenablestheefficientcalculationofcompactdoublefreeformsurfacesforcomplextargetirradiances. Thisis 1 demonstratedbyapplyingittothedesignofasingle-lensandatwo-lenssystem. 1 Keywords: Freeformdesign,Nonimagingoptics,Beamshaping,Beamprofiling. ] s c *[email protected] i t p o . 1 Introduction s c i s y In recent years the manufacturing of freeform surfaces has become increasingly feasible. These h p [ freeform surfaces offer an elegant way of simultaneous irradiance and phase control. Therefore, 1 v the development of numerical algorithms for the calculation of continuous freeform surfaces for 6 7 0 controloftheirradianceand/orthephaseofabeamisofgreatinterest. 3 0 . Inthisworktheproblemofdesigningcontinuousdoublefreeformsurfacesinageometricaloptics 1 0 7 approximation is considered, in which two collimated beams of arbitrary irradiance are mapped 1 : v onto each other. Several methods for phase and irradiance control with double freeform surfaces i X r a havebeenproposedinliterature. Oneofthefirstdesignmethodsforthemappingoftwowavefrontsbycoupledfreeformsurfacesis theSimultaneousMultipleSurface(SMS)method,whichwasdevelopedbyBenitezandMin˜ano.1 1 The surfaces are thereby constructed from generalized cartesian ovals and by applying constant optical path length (OPL) conditions.2 The design method can be utilized for numerous applica- tionsinimagingandnonimagingoptics.2,3 Zhang et al.4 and Shengqian et al.5 solve the design problem by describing it in the form of a Monge-Ampe`re type PDE, discretizing the equation by finite differences and then solving the re- sulting nonlinear equation system by the Newton method. The design method can be applied to a varietyofwavefrontshapes.5 Analternativeapproachtoconstructfreeformsurfacesforirradianceandphasecontrolisfromray mapping methods.12–23 These methods are based on the seperation of the design process into two separatesteps: thecalculationofanintegrableraymappingbetweenthesourceandthetargetirra- dianceandthesubsequentconstructionofthecontinuousfreeformsurfacesfromthemapping. The integrability thereby ensures the continuity of the freeform surfaces and the mapping of the input irradiance onto the ouptut irradiance. Since the integrability depends on the physical properties of theopticalsystemitisingeneralanontrivialtasktofindsuchamapping. As shown in several publications, there is a strong relation between the inverse problem of non- imaging optics and optimal mass transport (OMT).6–11 The cost function, which has to be applied to a certain optical configuration, is thereby problem-specific. For example, the mapping of two collimated beams with arbitrary irradiance onto each other with double freeform mirrors is de- scribed by a quadratic cost function7 and can be solved by corresponding numerical schemes.23 The same problem statement with double freeform lenses, which is considered in this work, is described by a different cost function, which depends on the OPL between the freeform surfaces asitwasshownbyRubinsteinandWolansky.9 The investigations presented here are inspired by several publications,13–18 in which the authors 2 applied the quadratic OMT cost function to calculate a ray mapping to deal with the lens design problem. With this ray mapping, designs have been demonstrated of both single freeform sur- facesforirradiancecontrolforcollimatedinputbeamsandpointsources,13,14,18 andthatofdouble freeform surfaces for irradiance and phase control.15–17 As demonstrated for illumination control with single freeform surfaces in Ref.21 and for collimated beam shaping with double freeform surfacesinRef.22 thedesignproblemsaretherebyrestrictedtoaparaxialapproximation. Here we first investigate the design by ray-mapping methods of double freeform surfaces which map two collimated beams with arbitrary irradiance onto each other beyond the paraxial approxi- mation. Toovercometherestrictiontotheparaxialregime,whichisnecessaryfortheconstruction of compact systems, the design problem will be modeled by two coupled PDE’s. This involves on one hand the Jacobian equation, expressing the local energy conservation, and on the other hand a ray mapping PDE, enforcing the surface continuity and the constant OPL. The PDEs will then be solved by an optimization scheme with the OMT mapping from the quadratic cost function as the intialiterate,leadingtoaconstructionapproachforthefreeformsurfaces. Todoso,theworkisstructuredasfollows. Insection2,byusingthelawofrefraction,theconstant OPLconditionandasurfacecontinuitycondition,aPDEforanintegrableraymapping,isderived. Together with the Jacobian equation it builds a system of PDEs for the determination of the map- ping components. It is argued that the PDE system is fulfilled within the paraxial approximation by the quadratic cost function OMT map. In section 3, a method for solving the PDE system for generallenslensdistancesispresented. ItisbasedondiscretizingthePDEswithfinitedifferences and solving the resulting system of nonlinear equations by a standard optimization scheme with thequadraticcostfunctionOMTmapastheinitialiterate. Asummaryofthedesignalgorithmand a detailed discussion of the implemenation is presented in section 4, followed by the application 3 of the presented method to the design of a single-lens and a two-lens system in section 5. Finally, insection6,ashortdiscussionoftheresultsispresented. 2 Freeformdesigninparaxialapproximation 2.1 Energyconservationandcostfunctions In Ref.21 a design method was presented for the construction of a single freeform surface for a collimated input beam with irradiance I (x,y) and an arbitrary illumination pattern I (x,y) S T on a target plane. It was shown that in the paraxial approximation the design process can be decoupled into two separate steps. In the first step a raymapping u(x,y) = (u (x,y),u (x,y)) is x y calculated from the theory of optimal mass transport, and in the second step the freeform surface isconstructedfromthemapping. There are several basic physical principles that a ray mapping needs to fulfill. Firstly, to map the source irradiance I (x,y) onto the target irradiance I (x,y), the ray mapping should be energy S T conserving. ThelocalenergyconservationisexpressedthroughtheJacobianEq. det(∇u(x,y))I (u(x,y)) = I (x,y). (1) T S Secondly,incaseoffreeformilluminationoptics,themappingshouldallowthecalculationofcon- tinuous freeform surfaces. As shown in several publications, these so called integrable ray map- pings are related to problem-specific cost functions representing different optical settings, where one has to distinguish between point sources and/or collimated beams, mirrors and/or lenses and soon.6–11 ThecostfunctiondefinesametricbetweenthesourcedistributionI (x,y)andthetarget S illumination pattern I (x,y) and therefore represents an additional constraint to the underdeter- T 4 mined Eq. (1). In the case of a single freeform surface for the redistribution of collimated input beams,thequadraticcostfunction (cid:90) d(I ,I )2 = inf |u(x)−x|2I (x)dx, (2) S T u∈M S which is valid in the paraxial approximation, was studied by the authors.21 A key property there wasthevanishingcurl ∂ u (x,y)−∂ u (x,y) = 0, (3) y x x y characterizingthequadraticcostfunctioninEq. (2).25 AsshownbyRubinsteinandWolansky,thecostfunctionforcollimatedbeamshapingwithdouble freeform lenses takes a different form than Eq. (2).9 The authors propose to minimize the corre- sponding cost function by a steepest descent algorithm to get the ray mapping,9 but unfortunatly a numericallystableimplementationisanontrivialproblem. Duetoitsapplicabilityintheparaxialapproximation(seebelow)andtheavailabilityofnumerous published stable numerical schemes for its calculation, the quadratic cost function OMT mapping serves as an initial iterate for the optimization scheme presented below. It will therefore build the basisofthedesignapproachpresentedinsections3and4. 2.2 Raymappingcondition We follow the approach from Ref.21,22 by expressing the basic geometry according to Fig. 1 in terms of the collimated input and output vector fields s and s , the refracted vector field s and 1 3 2 5 ∂ΩT ∂ΩT a) IT(x,y) b) IT(x,y) u(x,y) Id − z= z z= z T T s 3 z (x,y) II s4 s2 zI(x,y) s 1 z=0 z=0 (x,y) IS(x,y) IS(x,y) ∂ΩS ∂ΩS Fig1 a)TheirradiancedistributionsI (x,y)andI (x,y)withtheboundaries∂Ω and∂Ω aregivenontheplanes S T S T z =0andz =z ,respectively.Inthefirststepanintegrableraymappingu(x,y)=(u (x,y),u (x,y))iscalculated T x y betweenthedistributions,whichdefinesthevectorfields betweensourceandtargetplane. b)Inthesecondstepthe 4 freeformsurfacesz (x,y)andz (x,y)arecalculatedfromthelawofrefractionandtheconstantOPLcondition.The I II collimated in- and output beams are represented by the vector fields s and s and the refracted beam by the vector 1 3 fields . 2 theraymappingvectorfields : 4     0 u (x,y)−x    x          s1 =  0 , s2 =  uy(x,y)−y , (4)         z (x,y) z (u ,u )−z (x,y) I II x y I     0 u (x,y)−x    x          s3 =  0 , s4 = uy(x,y)−y.         z −z (u ,u ) z 1 T II x y T Since the goal is to calculate at least continuous freeform surfaces, we have to apply the surface continuitycondition n ·(∇×n ) = 0 (5) I I 6 tobothfreeformsurfacesz (x,y)andz (x,y)andtheirnormalvectorfieldsn (x,y)andn (x,y), I II I II respectively. Therebythenormalvectorfieldscanbeexpressedwiththelawofrefraction n = n ˆs −n ˆs , (6) I 1 1 2 2 intermsofthenormalizedincomingandrefractedvectorfieldsˆs andˆs andtherefractiveindices 1 2 n andn . HenceEq(5)canbewrittenas 1 2 {s ×[(s ∇)s ]} 2 2 2 3 s (∇×s ) = n −s (∇×s )+s (∇×s ) (7) 2 1 1 2 3 2 4 n·s 2 andtherefore,byplugginginthevectorfieldsgiveninEq. (4)anddefiningv := (u(x,y)−Id)⊥ = (−(u (x,y)−y),u (x,y)−x),weget y x (cid:2)(cid:0) (cid:1) (cid:3) v· v⊥ ·∇ v⊥ v∇z (x,y) = n · +v∇z (u ,u )−(z (u ,u )−z (x,y))∇v. (8) I 1 II x y II x y I n·s 2 Comparingthisequationwiththecaseofasinglefreeformsurface,21weseethatthev∇z (u ,u )- II x y termontherighthandside(RHS)arisesduetothesecondfreeformsurfacez (x,y)andistherefore II connectedtotherecollimationoftherefractedvectorfield. Thelefthandside(LHS)ofEq. (8)representsthedotproductoftheprojectedgradientofthefirst surface ∇z (x,y) =(∂ z (x,y),∂ z (x,y)) and the direction perpendicular to the ray deflection I x I y I (u(x,y)−Id). Therefore,anonvanishingRHSofEq. (8)contradictsthelawofrefraction,which states that the incoming, the refracted, and the normal vector have to lie in the same plane. This 7 canbeseendirectlybyusingtherelation     ∂ z (x,y) n2·(ux−x) ∇(z −z (x,y)) =! nI(x,y) ⇔  x I  =!  |s2|·(nI)z  ∝ v⊥, (9) I (n (x,y))     I z ∂ z (x,y) n2·(uy−y) y I |s2|·(nI)z leavinguswiththeconditionthattheRHSofEq. (8)hastobeequaltozero. ! AfurtherrelationcanbederivedbyapplyingthechainruletotheEq. ∇ z (u ,u ) = (∂ z (u ,u ),∂ z (u ,u )) = u II x y ux II x y ux II x y nII(ux,uy).24 Thisprovidesuswiththegradient∇z (x,y),whichisusedtorewritethesecondterm (nII)z II oftheRHSofEq. (8). Hencefromthecontinuitycondition(5)andthelawofrefraction(6)followthesystemofEqs. v∇z (x,y) = 0, (10a) I (cid:2)(cid:0) (cid:1) (cid:3) v· v⊥ ·∇ v⊥ g(u ,u ) x y n · +n −(z (u ,u )−z (x,y))∇v = 0, (10b) 1 2 II x y I n·s (n ) ·|s | 2 II z 2 withg(u ,u ) = −v2∂ u +v2∂ u +v v (∂ u −∂ u )andsimilarEqs. byconsideringEq. (5) x y x x y y y x x y x x y y andEq. (6)forthesecondfreeformsurfacez (x,y). II These Eq. (10) together with Eq. (1) build a system of PDEs for the unknown mapping u(x,y) andthesurfacesz (x,y)andz (x,y). I II To decouple the design process into separate steps as described in the beginning of this section, one needs to find a ray mapping fulfilling the condition (10b) which is nontrivial without any a prioriknowledgeaboutthefreeformsurfaces. Forsingleanddoublefreeformsthiscanbedoneby considering the small-angle approximation (z (u ,u )−z (x,y)) (cid:29) |u(x,y)−Id| leading to a II x y I vanishing first and second term in (10b). Addionally, using the mapping from the quadratic cost function defined by Eq. (3), the condition (10b) is fulfilled and the surfaces can be calculated by 8 usingEq. (10a)withappropriateboundaryconditions. Theboundaryconditionscanbederivedby considering the law of refraction (9) on the boundaries of I (x,y) and I (x,y).21 As discussed in S T Ref.21 ,thisleadstoapathindependentintegrationofEq. (9)tocalculatethesurfacez (x,y). I An alternative way to derive the validity of the quadratic cost function in the paraxial approxi- mation results is by utilizing an expansion of the Rubinstein-Wolansky cost function9 for small angles. ThedoublefreeformdesignprocesscanfurtherbesimplifiedbyalsousingtheconstantOPLcon- dition ! n |s |+n |s |+n |s | = const ≡ OPL. (11) 1 1 2 2 1 3 ByplugginginEq. (4),theEq. (11)canbesolvedfor n 1 (cid:113) z (u ,u )−z (x,y) = − OPL ∓ OPL2 +(n2 −1)|u(x,y)−x|2 (12) II x y I n2 −1 red n2 −1 red with n := n /n and OPL := (OPL − n z )/n between the first and second surface. The 1 2 red 1 T 2 sign in Eq. (12) depends on whether we have a single-lens (OPL > 0; n < 1: negative sign) red or a two-lens system (OPL < 0; n > 1: positive sign). According to Eq. (12), for single-lens red systemsthemappingvaluesarerestrictedbytherelation|u(x,y)−x|2 < OPL2 /|n2 −1|. red Since n · s and (n ) · |s | depend on z (u ,u ) − z (x,y), Eq. (10b) can be written as a PDE 2 II z 2 II x y I for the components of mapping function u(x,y) only. Therefore, the Eqs. (1) and (10b) build a system of PDEs for the functions u (x,y) and u (x,y). Both Eqs. build the basis of the design x y processfordoublefreeformsurfacesdescribedinsections3and4. 9 Beforewegiveanydetailswewanttodiscussthecondition(10b)brieflyforfreeformmirrors. 2.2.1 FreeformMirrors For mirrors the refractive indices in Eq. (6) have to be replaced by n ≡ n ≡ −1 and we get 1 2 n s = −(n )|s |. Therefore,Eq. (10b)reducesto I 2 II 2 (v2 +v2) x y ! − ∇v−(z (u ,u )−z (x,y))∇v = 0, (13) II x y I n ·s I 2 Hencethetwo-refractorproblemwithcollimatedbeamscanbesolvedifthemappingfulfills∇v = ∂ u −∂ u = 0, which is the case for the quadratic cost function defined by Eq. (1) and Eq. (3). y x x y ThiswasproveninamathematicallyrigorouswaybyGlimmandOliker.7 Hence, in contrast to the single lens, single mirror and double freeform lens systems, the design problem can be solved by the quadratic cost function without any additional assumptions like the paraxialapproximation. 3 FreeformLensdesignbeyondparaxialapproximation As mentioned in the previous section, the Eqs. (1) and (10b) are the basis of the design approach presented in the following. Since Eq. (10b) is exactly fulfilled by the mapping with the quadratic cost functions for an infinite distance between the freeform surfaces or an infinite OPL, respec- tively, the Eq. (10b) represents a correction to Eq. (3) for finite OPLs. Hence, for finite distances betweenz (x,y)andz (x,y)wearesearchingforcorrections∆u(x,y) = (∆u (x,y),∆u (x,y)) I II x y OPL→∞ with ∆u(x,y) → 0 to the ray mapping defined by the quadratic cost function, which we will denote by u∞(x,y) in the following. Hence, after writing Eq. (10b) in terms of Eq. (3) plus a 10