1、I. IntroductionD. K. Lynch and W. Livingston, Color and Light in Nature, (Cambridge university press, 1995), pp. 93-94. HYPERLINK /2014/07/architectural-caustics/ /2014/07/architectural-caustics/Input Beam InformationFreeformOptics?Output丨 Irradiance IInput Beam InformationFreeformOptics?Output Irra

2、diance & WavefrontEVOLUTIONARY ARTIFICIAL LIGHT SOURCESLASERS11111111111111111111111111*1A:A:11111111111111 1i1111117 、2 / 、21:-2uv一 + :Iin (U,巧 * e1(W1丿(叫丿- 1111LEDsY. Luo, Energy-saving LED light sour( SPIE NewsroomII. Formulation of the InverseProblemsAPPROXIMATIONSMonotonic Ray BendingGeometrica

3、l OpticsPoint Light SourceLossless SystemSOME FORMULATION REFERENCESJ. S. Schruben, Formulation of a reflector-design problem for a lighting fixture,J. Opt. Soc. Am. 62, 1498-1501 (1972).IR. Winston, J. C. Minano, and P. Benitez, eds., Nonimaging Optics (Elsevier, 2005), PP. 174-178.IIH. Ries and J.

4、 Muschaweck, Tailored freeform optical surfaces, JOSAA 19(3), 590-595, (2002)IH. Ries, Laser beam shaping by double tailoring, Proc. SPIE 5876, 587607, (2005)IIWu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, Freeform illumination design: a nonlinear boundary problemIfor the elliptic Monge-

5、Ampere equation, Opt. Lett. 38, 229-231 (2013).IIY. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, Double freeform surfaces design for laser beam shaping withIMonge-Ampere equation method, Opt. Commun. 331,297-305 (2014).IIS. Chang, R. Wu, A. Li and Z. Zheng, Design beam shapers with double free

6、form surfaces to form a desiredIwavefront with prescribed illumination pattern by solving a Monge-Ampere type equationJ. Opt. 18, 125602 (2016).NUMBER OF FREEFORM SURFACESIrradiance ControlAt least two freeform surfacesIf we use two freeform optical surfaces to con trol the irradia neeon ly, we must

7、 provide an additi onal con stra int.EXAMPLE: DOUBLE FREEFORM SURFACE DESIGNFOR IRRADIANCE & WAVEFRONT CONTROLEn ergy Con servati on8u dv du dv Ray Trac ing Equati onsHQ = P + P, QR = P + rR 2.|W = Q + Q,WO = Q + tO | 廠-瀘-曲 + 卜護(hù)亠-N)TLII1qS, P + 空P, Q + ng W =ConstSurface Normals and ContinuityN=（匕 x

8、 p）/|匕 x 瑞| a2 pa2p |=O = （%xwj/lwxwj| audvdvdu |Parametric surface normalsSurface continuity17VERY COMPLICATED DERIVATION PROCESSRay Trac ing Equati onsEn ergy Con servati onParametric Surface NormalsIo(u, v)=厶 3)Surface Con ti nuityd2 pd2 pdu dvdvduOPL Con sta ncyt = g(u, v, P, Pu , Pv ) 耳=測,v, P,

9、 Pu , Pv )SECOND-ORDER NONLINEAR PDEL 學(xué) 學(xué)-(空)2 + B 學(xué) + 2C 蟲 + dD + oli du u dv v du dvdu2 du dv dv2ii一LjBoundary condition:,v,pPu,幾)：o0 昭= (u, v, P, Pu , Pv )A = A(u, v, P, Pu , Pv )b = b (u, v, p, pu , pv ) Monge-Ampere equation c = c(u,v,p,Pu,Pv) with tedious coefficientsD = D (u, v, P, Pu, Pv)E =

10、 E(u, v, p, Pu，Pv)DIRECT DETERMINATION METHODSH. Ries & J. Muschaweck, JOSA A 19, 590-595,(2002)H. Ries, Proc. SPIE 5876, 587607 (2005) TOC o 1-5 h z IIIIIIIINumerical technique: Multi-grid algorithmR. Wu, et al., Opt. Lett. 38, 229-231 (2013).IIY. Zhang, et al., Opt. Commun. 331, 297-305 (2014).III

11、INumerical technique: Newtons methodII2p_；III. Simplified Design MethodsSimplifieddesig nmethodsSupport ingquadricmethodsLin earprogram mingmethodsParametricoptimizati onmethodsRay mapp ingmethodsivDetaileddescription hereSUPPORTING QUADRIC METHODSV I. Oliker, Mathematical aspects of design of beam

12、shaping surfaces in geometrical optics, Trends in Nonlinear Analysis, pp. 191-222 (2002).F. R. Fournier, et al., Fast freeform reflector generation using source-target maps, Opt. Express 18, 5295-5304 (2010).D. Michaelis, et al., Cartesian oval representation of freeform optics in illumination syste

13、ms, Opt. Lett. 36, 918-920 (2011)S. Magarill, Skew-faceted elliptical reflector, Opt. Lett. 36, 532-533 (2011).L. L. Doskolovich, et al., Design of mirrors for generating prescribed continuous illuminance distributions on the basis of the supporting quadric method, Appl. Opt. 55, 687-695 (2016)V Oli

14、ker, Controlling light with freeform multifocal lens designed with supporting quadric method(SQM), Opt. Express 25, A58-A72 (2017).I1Quadrics: Cartesian ovals, ellipsoid, paraboloid, and hyperboloidiLINEAR PROGRAMMING METHODST. Glimm and N. Henscheid, ISRN AppliedMathematics, 2013,635263 (2013).Tmax

15、 c x, s.t. Ax bT. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem”，J. of Mathematical Sciences, 117(3), 4096-4108 (2003).Xu-Jia Wang, “On the design of a reflector antenna II,” Calc. Var. 20, 329-341 (2004).V. Oliker, “Geometric and va

16、riational methods in optical design of reflecting surfaces with prescribed irradiance properties”, Proc. SPIE 5942, 594207 (2005).T. Glimm a nd N. Hen scheid. Iterative Scheme for Solvi ng Optimal Tran sportation Problems Arisi ng in Reflector Desig n. ISRN Applied Mathematics, 2013,635263 (2013).C.

17、 Can avesi, W. J. Cassarly, and J. P. Rolla nd. Observati ons on the lin ear program ming formulati on of the sin gle reflector desig n problem. Opt. Express 20,4050-4055 (2012)PARAMATRIC OPTIMIZATION METHODSmin f (v), s.t. v e KK is the feasible region二工 v,jxiyji+j NPablo Ben itez and Juan C. Mina

18、no, The Future of Illumi natio n Desig n, Optics & Photo nics News 18(5), 20-25 (2007).乙 Liu, P. Liu, and F. Yu, Parametric optimization method for the design of high-efficiency freeform illumination system with a LED source.Chin. Opt. Lett.10: 112201-112201 (2012).RAY MAPPING METHODSVARIABLE SEPARA

19、BLE RAY MAPI0(u, v ) dudv = /i(f,)dfd【0,u (Uo,v (V)dudv = L(“)坤耳 = l(u),耳IIW. A. Parky n, “l(fā)llumi nati on len ses desig ned by extrin sic differe ntial geometry”，SPIE 3482, 389-396 (1998).D. L. Shealy and S. Chao, “Geometric optics-based design of laser beam shapers,” Opt. Eng. 42, 3123-3138 (2003).

20、L. Wang, K. Qia n and Y. Luo, “Disco ntinu ous free-form lens desig n for prescribed irradia nee”，Appl. Opt. 46, 3716-3723(2007).Y Ding, X. Liu,乙 R. Zhe ng, and P. F. Gu, “Freeform LED lens for un iform illumi natio n,” Opt. Express 16, 12958-12966(2008).COMPOSITE RAY MAPCxd,ydtH)D. Ma, Z. Feng, and

21、 R. Liang, Freeform illumination lens design using composite ray mapping, Appl. Opt. 54, 498-503 (2015)POLAR-GRIDS MAPg Vk)刖,如J-(0$+l 9 卩At) I (仇+1加,如Ilf(仇/+/,卩人)& - emax(Ps+ik 1, %+l)/(M+f A+l)X. Mao, H. Li, Y. Han, and Y. Luo, Polar-grids based source-target mapp ing con struct ion method for desi

22、g ning freeform illumination system for a lighting target with arbitrary shape, Opt. Express 23, 4313-4328 (2015)min jj I0(u,v)c(u,v,g,q)dudvs.t I0(u,v)dudv 二厶dgdq :J. Rubinstein and G. Wolansky, Intensity control with a free-form lens, J. Opt. Soc. Am. A 24, 463-469 (2007).A. Bruneton, A. Bauerle,

23、P. Loosen, and R. Wester, Freeform lens for an efficient wall washer, Proc. SPIE 8167, 816707 (2011).A. Bauerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, Algorithm for irradiance tailoring using multiple freeform optical surfaces, Opt. Express 20, 14477-14485 (2012).A. Brun eto n, A. B

24、auerle, R. Wester, J. Stolle nwerk, and P. Loose n, High resolution irradia nce tailori ng usi ng multiple freeform surfaces, Opt. Express 21, 10563-10571 (2013).Z. Feng, L. Huang, G. Jin, and M. Gong, Designing double freeform optical surfaces for controlling both irradiance and wavefront, Opt. Exp

25、ress 21,28693-28701 (2013).Z. Feng, B. D. Froese, and R. Lia ng, Freeform illu min ati on optics con structi on follow ing an optimal tran sport map, Appl. Opt. 55, 4301-4306 (2016).30C. Bosel and H. Gross, Ray mapping approach for the efficient design of continuous freeform surfaces, Opt. Express 2

26、4, 14271-14282 (2016).c (u, v,帥=1 (f - u )2 + ( v )2 Io(u, v) 厶(W)Standard Monge-Ampere EquationR. T. Rockafellar, “Characterization of the subdiffere ntials of con vex fun ctio ns,” Pacific J. Math. 17, 497-510 (1966).Robert J. McCa nn，“Existe nee and uniquen ess of monotone measure-preserv ing map

27、s,” Duke Math. J., 80, 309-323 (1995).J. D. Benamou, B. D. Froese, and A. M. Oberman, “Numerical solution of the optimal ransportation problem using the Monge-Ampere equati on,” J. Comput. Phys. 260, 107-126 (2014).SURFACE CONSTRUCTIONPoi nt-by-poi ntW. A. Parkyn, Illumination lenses designed by ext

28、rinsic differential geometry, SPIE 3482, 389-396 (1998).L. Wang, K. Qian and Y Luo, Discontinuous free-form lens design for prescribed irradiance, Appl. Opt. 46, 3716-3723 (2007).Z. Feng, L. Huang, G. Jin, and M. Gong, Designing double freeform optical surfaces for controlling both irradiance and wa

29、vefront, Opt. Express 21,28693-28701 (2013).First-order PDED. L. Shealy and S. Chao, Geometric optics-based design of laser beam shapers, Opt. Eng. 42, 3123-3138 (2003).Y Ding, X. Liu,乙 R. Zheng, and P. F. Gu, Freeform LED lens for uniform illumination, Opt. Express 16, 1295812966 (2008).Least squar

30、esA. Bruneton, A. Bauerle, P. Loosen, and R. Wester, Freeform lens for an efficient wall washer, Proc. SPIE 8167, 816707 (2011).A. Bauerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, Algorithm for irradiance tailoring using multiple freeform optical surfaces, Opt. Express 20, 14477-14485

31、 (2012).A. Bruneton, A. Bauerle, R. Wester, J. Stollenwerk, and P. Loosen, High resolution irradiance tailoring using multiple freeform surfaces, Opt. Express 21, 10563-10571 (2013).Z. Feng, B. D. Froese, and R. Liang, Freeform illumination optics construction following an optimal transport map, App

32、l. Opt. 55, 43014306 (2016)33Step 1: Define an in put ray seque nee and output rayseque nee based on the optimal transport mapStep 2: Give n an estimate of the first freeform surfaceStep 3: Con struct the sec ond freeform surfacebased on the OPL con sta ncyStep 5: Compute a no rmal field based on Sn

33、 ells law39Retur n to Step 3, and repeat the above processNI, j+1N (VxN)豐 0I. /八I 1 i+1, J+11 i+1, JN .J Ni+1, j+1I汁 1,j+1I (Pi, j+1 - P j)(N i, j+1+N i, j)(P+1, J- P j)(N i+1, j + N i, j)0L1LEAST SQUARES SOLUTION Zexin Feng, Brittany D. Froese, and Rongguang Liang, Freeform illumination lens constr

34、uction following an optimal transport map, Appl. Opt. 55, 4301-4306 (2016)EXAMPLECen tral Thick ness - 15 mm Target Dista nee = 1000 mmZex in Feng, Br itta ny D. Froese, and Ron ggua ng Lia ng, Fr eeform illumi natio n lens eon struct ion followi ng an optimal tran sport map, Appl. Opt. 55, 4301-430

35、6 (2016)IV. Design Methods Under Paraxial and ThinLens ApproximationsGEOMETRIC OPTICS METHODIin(x, y)I:(x, y, w)(乙 n，d)1Fermats prin ciple or the method of thestati onary phase:； ；dwg xdw葉一 yI I:dxd dyd F. M. Dickey and H. C. Holswade, Laser Beam Shap ing: Theory and Tech niq ues (Marcel Dekker, 200

36、0).O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092-1099 (1974).A SIMPLE DERIVATION PROCESSFermats PrincipleEnergy Conservation：dwx11 dw一 ydxd 11dyd1T1-T dwdwf x d ,:dx1L y ddy1 F. M. Dickey and H. C. Holswade, Laser Beam Shap ing: Theory and Tech niq ues (Marcel Dekk

37、er, 2000).111io(u, v)m)11111 .df d葉 d葉df du dv du dv+Surface Continuityd2 wd2 wdxdydy dx45THE RESULTING EQUATIONStandard Monge-Ampere EquationA MUCH SIMPLER CASE:VARIABLE SPARABLEVariable Separati onWavefront Integration1 一 1 一- -一一 一 一亦一5X亦廠 =x), = y)I0( x, y ) dxdy = I1(,) dgdqO. Bryngdahl, “Geome

38、trical transformations in optics,” J. Opt. Soc. Am. 64, 1092-1099 (1974).K. Nemoto, et al., , “Laser beam-forming by deformable mirror,” Proc. SPIE 2119, 155-161 (1994).Y. Arieli, N. Eise nberg, A. Lewis, and I. Glaser, “Geometrical tran sformati on approach to optical two-dime nsional beam shap in

39、g,” Appl. Opt. 36, 9129-9131 (1997).H. Aagedal, M. Schmid, S. Egner, J. Muller-Quade, T. Beth, and F. Wyrowski, “ An alytical beam shap ing with applicati on to laser-diode arrays,” J. Opt. Soc. Am. A 14, 1549-1553 (1997)Z. Zeng, N. Ling, and W. Jia ng, “The in vestigati on of con trolli ng laser fo

40、cal profile by deformable mirror and wave-fro nt sen sor,”Jour nal of Modern Optics 46, 341-348 (1999).DIRECT CACULATION OF THE FREEFORMSURFACE FROM WAVEFRONTFreeform MirrorIFreeform Lens TOC o 1-5 h z I：|UZ=z(x,y)I I；z=0 z=HWlens = -nZ(X, y) - H - Z(x, y)IIJ. W. Goodman, Introduction to Fourier Opt

41、ics, 2nd ed. (McGraw-Hill, 1996)CONSIDERING DIFFRACTIONi存孑+心e 2 di加i- ( x? + y 2)FT E(x, y)e 2dIterative Fourier Transform Algorithmsekd i乂 (嚴(yán) +滬)亠(x2 + y2)= - e2 dFT E (x, y )e 2 diidiiForward Fourier tran sformE (x, y) = yj Io( x, y) e_ _Rema in Remain 如 E(仙)=In verse Fourier tran sformE (x, y)=害寺

42、(+ 尸)FT-1d-止(孕+心En)e 2dGerchberg-Saxt on (GS) algorithmR. W. Gerchberg and W. O. Saxton，“A practical algorithm for the determ in ati on of phase from image and diffracti on pla ne pictures,” Optik 35, 237-246 (1972).J. R. Fie nup, “Iterative method applied to image recon structi on and to computer-g

43、e nerated holograms,” Opt. Eng. 19, 193297 (1980).O. Ripoll, V. Kettu nen, and H. P. Herzig, “Review of iterative Fourier tran sform algorithms for beam shapi ng applicatio ns,” Opt. Eng. 43, 2549-2556 (2004).C. Bechet, A. Guesalaga, B. Neichel, et al., “Beam shaping for laser-based adaptive optics

44、in astronomy,” Opt. Express 22, 12994-13013 (2014) .50COMPOSITE METHODSGeometries methodsInitial phase estimateIterative Fourier Tran sform AlgorithmsGS or variants of GSM. T. Eismann, A. M. Tai, and J. N. Cederquist, Iterative design of a holographic beamformer, Appl. Opt. 28, 2641-2650 (1989).X. T

45、an, B. Gu, G. Yang, and B. Dong, “Diffractive phase elements for beam shaping: a new design method,” Appl. Opt. 34, 1314-1320 (1995).X. Deng, D. Fan, Y Qiu, and Y Li, “Pure-phase plates for superGaussian focal-plane irradianee profile generations of extremely high order,” Opt. Lett. 21, 1963-1965 (1

46、996).J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. 27,1463-1465 (2002).乙 Feng, et al., “A composite method for high resolution freeform optical beam shapi ng,” Appl. Opt. 54, 9364-9369 (2015).EXAMPLE52乙 Feng, et

47、al., “A composite method for high resolution freeform optical beam shapi ng,” Appl. Opt. 54, 9364-9369 (2015).10.24mmSIMULATION RESULTSComposite methodGeometric optics methodGS53Z. Feng, et al., “A composite method for high resolution freeform optical beam shaping,” Appl. Opt. 54, 9364-9369 (2015).V

48、. Conclusions & OutlookCONCLUSIONIFreeform optics design for lllumination & Beam Shap ing is a very difficult in verse problemtA Q2p Q2p _( 82p Qu2 8v2 Qu8v)2 + B 竺 + 2C 沁 + D 遼 + E = o| Qu2QuQvQv2CONCLUSION:IAside from the direct determ in ati on methods, we can use simplified design methods eg, ray mapping methods1i JiVCONCLUSIONThe design problem under paraxial and thin lensIapproximations