ThammasatInt. J. Sc.Tech., Vol. 9, No. 2, April-June2004

Lie AtgebraicMethods of Light Opticsfor Lens SystemDesign used in OTIS Architecture

W. Pijitrojana Departmentof Electrical Engineering,Faculty of Engineering ThammasatUniversity, Klong Luang, Pathum Thani l2l2l, Thailand Phone0-2564-3001-9 Ext.3045, Fax0-2564-3010,E-mail: [email protected]

Abstract The use of Lie algebraic Methods offers the calculation of high order aberrationsand also new insight into the origin and possible correction of aberrations.In this paper, it will be shown using Lie algebraicmethods, how to describethe transpositionof the OTIS, characteriseits optical elementsand simpliff the calculation of their aberrations.A preliminary numerical study of its optical elements indicatis that, by simultaneouslyremoving almost all third-order (and higher-order) aberrations,it is possibleto design a lens system,a Fourier transform lens, having an eror function approachingzero.

Keywords: Lie algebraicmethod, Lie Optics,Optical interconnection.

1. Introduction In thin lens (axis-symmetry) system as The propagation of a ray in an optical shown in Figure l, the objectphase space(q',p') system can be described in many cases by a in geometrical optics is transformed canonically Hamiltonian. The Hamiltonian formulation of to the correspondingimage space(q',p') through geometrical optics describeslight rays as points its representationmatrix which is the composite (q, p) in an optical phase space' The position ofthe operatorsofparaxial free propagation(q of a ray g=lq,,q,) determinesthe coordinate and ,n,) in a homogeneousmedium of refractive The intersection with the 0 Plane. index r after the object and before the screen,by p =(p,,pr) is the momentum coordinate distances z, &nd zr, arnd the action of the projection, on the plane, of the ray direction refraction surfaces (q). The concatenation of vector. Fermat's principle leads to an optical these operators produces the Lie map of the Lagrangian [1,2,3] from which the canonical system % =r$,F2. These operators are Lie momentum p is shown to be a vector in the z : are given by constantplane, along the projection ofthe ray on (exponential)operators, and they the plane,of magnitudep = nsin? wheren is the , - z,lnr)l refractive index of the medium at (q, z), and d is ='{''-+'[l o,=*,[?(-,. r )) between the ray and the z axis. The *?',',)] the angle 0)l initial and final values of the system are n,'= *ol-;(': '')*;o,uuo,l =n{or,[rt, representedby ni and wr, respectively.The set t)l of vectors p is the optical phase space.The ','= *ol?(-'.;t',t)] = - z,/rl)l initial conditions determine the final conditions 4''-r'[lr)) and can be expressedin terms of a functional (1) relationship that is denoted formally as the optical transfer map or optical symplectic map. where L = 2l zz and n' is the refractive index of The optical symplectic map can be written as a thin lens. Let z, = z, = (focal length of thin product of Lie transformations. The Lie 7 (air) : l, and The transformation is a linear operator acting on lens), n a=-/r. phase space, and is formally functions of transformation lvt of the thin lens system is series. defined by the exponential given by

48 ThammasatInt. J. Sc. Tech.,Vol. 9, No. 2, April-June2004

(b) Refractionat the boundarysurface between v e]o-;loI r rr rt,l I r I otl I r I "rrt,l two regionsof refractiveindices ,, &11dn2 , ' llq]oo'l'rr 'llc1o'r,{o i the formula is: -.10.r.l o /1]'f -t 7" \ llfr orl fq,) ( t o)fq) = "*p( ,)"*p(t-,')lQ,),q' ')"^p( :| t-t il l 7 1z),p' f lz):p' r)\nt) (2) \nt) [-p ( I o) , n) n, detl ,,-l.P= ^ (6) where r is the radius of thin lens. The Lie \-p t) ^ operators: lr: is definedas: where R is the radius of the refracting )/r'on o o.\ :i,: :=Il -on l,=h..1 (3) surface. Alaq, dP, ?P,aq, ) From these two results. we can calculate the matrix of the thin lens (a double convex lens) between a reference planes r, located a distance r) /(a focal length of the thin lens) to the left of the lens and a reference plane r, located a distance/ to the right (Figure 1), then we have: (';,)=lr il[l) (7,

whereI u.,,{L fl .* istheradiusofthe I \R, Rt) Figure 1. Thin lens system. left refracting surfaceand n, is the radius of the right refracting surface. From equation (7), the elementsA and D of the matrix are zeros. This The matrix 1 o 1r1from equation (2) is l-r/rr ol meansthere are two caseswe will consideras: corresponding to the transformation matrix in 1) D = 0. This meansthat all rays enteringthe geometricaloptics [5], i.e.: input plane from the same point emergeat the output plane making the same angle with the e a1(a,) r+r axis, no matter at what angle the rays enter the ln,l=l ,fr,l system.In anotherwords, the positionof the ray Ir./ \c D)\P,) lP'/ is transformedinto an angle. 2) A = 0. This means that all rays entering the system at the where M &re 2x2 matrices of determinant 1. sameangle will passthrough the samepoint in These kinds of matrices are called (linear) the output plane. Therefore, the angle is canonical transformations or symplectic transformedinto the position of the ray With transformations.Any refractinglens systemcan [6]. two cases it shows that the lens system be considered as the composite of several transforms the coordinate system ,p by systemsof two basictypes: \q ) (a) Afer translation by a distance t, the formula rotatingthe axes by e0' : is: (n,r)n (-o,n) (8) ( q,\ (l 'l[n I t=l ] lp,/ \o r )\P,) In this paper,we usedLie algebraicmethods /r r ) to describe the system called the Optical detl I t r t/ /n (5) (OTIS). l0 t) TransposeInterconnection System The OTIS system,which is describedin Ref. 4, has threestages of lenses,an arrayof microlensesin where n is the refractive index of the I, macrolens in stageII, and an array of medium. stage a microlensesin stageIII, as shownin Figure2.

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2. Fundamental Theory of the OTIS -T\,\ The transposeinterconnection ofthe system (o) _[' is a one-to-onemapping between u"u inputto \P ),. o (10) lv x rl output beamlets. The input and output lh fo),. beamlets are arranged as an .lrv,.[u array of .[n,,[u sub-array. Each .[i",[u sub-array of ( Af, \ input and output beamlets are at the front and (q\ -l--Tol back focal planes of each of the lensesof stageI (r \p),.-| 4o I r) and III which are arrangedas an ,[i",[u anay, l. /t/: )-o respectively.Each input and output beamletshas a coordinate (n,n\ where n,n=r,...,,[u",[u. The whete fr,fr, and ft are the focal lengths of input beamlet with the coordinate (n,r) is lensesin stageI, II, and III, respectively,and the mapped to the output beamlet with the effective focal length of the system is ny coordinate (n,n), called the transposition of the ff. input. We can write as the equation: Gaussian beam analysis and linear optical desigrr, the optical transpose interconnection (n,n) - (n,n) (9) system [8] has been designed and simulated as shown in Figure 3. The sources of input beamlets are telecentric sources, i.e. the chief ray of each source is 3. Aberrations in the OTIS parallel to the optical axis of the corresponding Let *t be any symplectic map having a lens in stage L The microlenses in stage I are Taylor seriesexpansion of the form, positioned one focal length away from the sources. The macrolens in stage II and z{ = K, +ln.rz; *7r,*z'rz: microlenses array in stage III are positioned as 1tZ) shown in Figure 2(a) so that the images of the +ltJ "o"ot'ut'"to* ,.. input beamletscan be Fourier transformedat the output. Figure 2(b) shows the diagram of the system for N and M = 16. The system An incoming initial ray is specified by the compositesof 256 input and output beamletsare phase-spacecoordinates z', and the outgoing arranged&s 8rr 4x4 array of 4x4 sub-array, an final ray has the corresponding phase-space 4x4 dfid! of lenses in stage I, a macrolens in coordinates zr . From equation (12), it can be stageII, and an 4x4 srrd! oflenses in stageIII. shown that 9! can be written uniquely in the The input beamlet with coordinate (r,r), for factor product form, example, on the microlens n : I in stage I is mapped to the output beamlet with coordinate ctil= exp("4 )"*p( l, :)exp( l, :)exp( I )... : (r,r)on the microlens n I in stageIII, similarly exp(.f^:) (ts) the input beamlet with coordinate (1,2)on the -- microlens n | in stage I is mapped to the where each function /) is a homogeneous output beamlet with coordinate (z,r) on the polynomial of degreem in the variablesz' .By microlensn = 2 in stageIII, and so on. Hence, definition, knowledge of W is equivalent to the hansposition of the input coordinatesto the knowledge of the relation between the initial output coordinatescan be done by three lenses. conditions z' and the final conditions zt . Therefore, the optical transposeinterconnection According to equation (13), knowledge of system composite of three stages of lenses to W amounts to transform the input beamlets with coordinates determining certain homogeneous polynomials ,f,,.fr,.fr.fn,etc. In light optics,the into the coordinatesof the outputbeamlets I i I factorization theorem indicates that the effect of any collection of elements can be characterized lo^l . Equation (10) below is the equationof by a set of homogeneouspolynomials. It can be the system: shown that the polynomial f, reproduces the

50 ThammasatInt. J. Sc.Tech., Vol. 9, No. 2, April-June2004

constantterms K,, in the Taylor expansion,and the polynomial /, reproducesthe linear matrix (paraxial optics) terms R", . And the polynomials f., f^,etc. describe departures from paraxialoptics and reproducethe nonlinear terms f r. ,U ,,n , etc.Moreover, unlike the terms K., R.n,T,,b,,etc.,the polynomials 1,, are all independent.Much of geometricallight opticsis concerned with systems composed of lenses having axial symmetry and constantindex of refraction.For suchsystems the transfermap M can be written in the form Coordinatesof lensesand inputsin stageI

M = exp(:/, :)exp(:y* :)exp(y. :)... (14)

Only f _ in equation(13) with even lz occurs due to symmetry,and it can dependonly on the variablespt,q',andp.q. From equation(14), the resultsof the equation(2) is one part of an expressionof the form

(l5)

The quantity M(j denotes the Gaussian (paraxial)porrion of the map r exp( :) 7, r and Coordinatesof lensesand outputsin stage is given by the relation of the equation(2). The III last term, exp(:/rj :), describes third order (b) Front View of inputs,outputs, and departure(aberrations) from Caussianoptics. [n mesolensesin stageI and IIL general,l; hasan expressionofthe form Figure2. Diagramof the OTIS with N = M = 16 t7l. l,i : A'(p'I .q)+ .q)' .qh' + B'p:(p c"(p * D'p'q' + r'(p * r'(q, )' (l6)

(a) SideView of layourof the OTIS Figure 3. Modelling of the OTIS [7]

5l ThammasatInt. J. Sc. Tech., Vol. 9, No. 2, April-June 2004

The Seidel third order aberrationsare classified Microlensesin stage.Ill p;,= p. = -(,' - t),ebl,-(,' - t)lr,fu',, + fit, [9,10,1l] and given in Table I below [r qi,,= q"' =(r *",),e -b'-t)tlb',,+l(r,-(n -lyyy.)1',fu|,+fil', Coefficient Term Seidel Aberration I tV tp'I Sphericalaberration Spherical aberration (t'): B' p'(p'q) Coma (18) C' (p'q)' Astiematism ,(l) = [,"_oo;.,,o.) D p'q' Curvatureof Field

D b'qh' Distortion This aberration unfocuses qc, the ray position F' (q,I Pocus with translation, but produces no change in the ray direction with translation. Table 1. The Seidelaberrations. Coma (n'): The determination of the function 4 and its carried out using the coefficients can be po+B'p'po (tsl Campbell-Baker-Hausdorff formula [8]. Each "'[qJ^,(p\_( - ,l lq" - B'Pp' qpo* p'qo!) lens of the system can be expressedas a linear part and third-order terms, which are nonlinear a cone (lpol .onttunt) issuing from (cubic) transformations of phase space, i.e., Here rays on aberrations. There are also inhomogeneous an object point qo fall on a circle with center at terms in the system becausethe translations of This o,=qo(t-n'nz) and radius p=r'p'lq'l' the optical axis of the microlensesin stageI and "comet" gives to the familiar 60o image of III. To examine the effect on phase space of rise two meridional rays on the cone fall each of the aberration coefficients points. The the samepoint in the direction of qo, and the (A', B',c', D', n', r') and translation,the cubic on monomials, which are produced out of p and q two sagittal rays (perpendicular to the former) through the action of the operators of the on the diametrical opposed point, also in the system, are considered. These monomials are direction of gc. Momentum space pc is also q'q' To considerthe p'p, ptq, P'

fn')_rfn)_[n"] (r7) Due to the common p.q coeffrcient, sagittal \q'l (q/ \qol rays are unaffected but meridional ones are. Rays on a cone map onto straight segments, This translation produces change in the ray centered and directed along qo, of half-length directionand the position. 2c'q'lpol.

Microlensesin staeeI -lQ'-1)1r,fo+zt, offield p'- p" = [t-("'-l)f h Curvature (o'): = = (r * - ("'- rM' * ( r,- (n'- lfi)l r,fu+ i t' ei co, I ).n b (2r) Macrolensin stageII ,[;)=[t:li"::i:) t'* tr = lr- (" - t.ff/..,fo',- l(n'- t)l',fo',- i t" Qrr=Q6u=10*n)f,-("-t)tib',+l(r,-(n'-t[)l'rfo],-it, positionsare unfocused as well asmomenta of the phasespace origin. Rays on a conemap onto

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a circle with center qo and radius 2o'4'lpol. where y,,y, and 7 are the focal lengths of the They fall into focus in qc on a paraboloid lensesin stageI, II, and III, respectively.r,,rr, z =2D'q2. and r3arethe radii ofthe lensesin stageI, II, and III, respectively. ,' is the refractive index of the lens. and or€ Distortion (E )i n=\...,JN fi=;..,Ju. A,andA, the pitches of the lenses in stage I and III, El2n.rr.o*q'poll respectively. "'[qJ.,(p\ -[_(p"+ e2) qo-E'q'qo ) 4. Discussions This is the Fourier conjugate of coma. Position l. Equations (17)-(23) show that sphericaL two-space is distorted (r' > 0 it is of the aberration and coma are the dominant "barrel" type), but it remains in focus, while in imperfections on phase sphce.It also shows that momentum space the comatic phenomenon the macrolens in stage II is the most critical appears. component in the system because the focal length and the radius of the macrolensare large. Pocus (r'): 2. The diameters of microlenses and a macrolenscan be scaled down to t.gzJN and 19.9il^N respectively.The focal lengthsof +aF'a'qol Q3) [8], *(v\=(n" microlenses and a macrolens are also scaled \q/[ Qo ) down to t.u.Ju and 9.992J1,1, respectively. This This is an pc-unfocusing aberration, Fourier implies that the radii of the microlensesand the macrolens can be also scaled down to the conjugateto spherical aberration.Position space magnitude of wavelength. This means that the is neither distorted nor unfocused.so this has not effects on phase space of each aberration been usually counted among the Seidel coefficient can be scaled down to the magnitude aberrationsof the system. For F' >0, ray slopes of the propagatory wavelength. By using the increase cubically with in the direction of lqol design of optical MEMS with many optical qo and thus depth of field decreasesaway from codes such as GLADTM. microlenses and the origin. macrolensesfor the size of wavelength can be The aberration coefficients in equation (16) are fabricated. given by 3. The third-order terms in each equation represent aberration with Seidel aberration 1' = - y I t + [(n'- r\zn' + r)/t n'rly' - l(n'- r\2,'- r)l+,' lr' coefficients A',8' ,c' ,D' ,E , and F' . These *l(n'-r\+n'' -t,' *+lv'11'-11,' -rf lu'11' Q4) aberration coeffrcients are derived from the intrinsic aberration coefficients, i.e., B'= -lb" - r)lz"',1 *lz,' 1n' - r)f zr'11' A,B,C,D,E,F, plus the transition from the -l(,'-tft;, - s,'*z)lz,'ly' +\fu' -rf lztll' Q5) front focal point to the lens and the transition from the lens to the back focal point [8]. From c' =-[(," -rlz,'l +l(n'-r\2,''-tn'*zlz,'l' -1fu'-rf lz,'j11' theoreticalpoints of view, therefore,the intrinsic (26) aberrationsofa single lens can be correctedby a well-designed compound lens system but there o' =(n' - | I an' r - l(""- | 1+,' 11 * lfu- r\z"'' - t "' +z)l +,' l1t' are still imperfections from the transition and misalignments. The translation terms depend on the -[,'-rl l+']1' (27) 4. positions of the microlenses in stage I and III, i.e., the term and r' =(n'-r)fzr'-[("'-t\,'' -"'+tyz,'lf +l(,'_r)o f z,o];' 7tr, ;76,. (28) 5. The macrolens in stageII is a bi-convex performs r' = n'(n'-t)/srl (2e) single lens, which as a Fourier transform lens. From those equations, the aberrationcoefficients ofeach lens in stageI, II, and III can be obtained as:

53 ThammasatInt. J. Sc. Tech.,Vol. 9, No. 2, April-June2004

Microlensesin stageI (Unit: mm.) where 6, and c. &re the surface A' = 5.46 B' - 0.162 curvaturesof a thin lens. x can also be written C- = -0.001 D'-O.0O2 in terms of the lens power, K, and refractive a p'=4.7Jx10-6 E = 1.9'l xl} indexof a thin lens,n', as Macrolensin staseII A' = 30.315 B' = 0.33 .. n l, , ( x - 1.,+(,, 31) C- = -0.001 D'=4.48x104 E-=-0.6x10 s F. = 2.5-5x l0 8 Microlensesin staseIII The following relationsare alsoeasily proven: A' = 5.16 B' = Q.t62 c'- 0.001 D'-O.OO2 r r(x+r) 6 'l E - 1.9'7xl}a F. - 4.75xlO r 2(n'-t) (32) 1 ,(a(X- 1) ' From these results, spherical aberration and \ 2\n'-t) coma are the dominant imperfections. This showsthat the macrolensof the OTIS systemis The shape factor is also often called bending the most critical componentwith respectto these factor. If we substitutethese equations into the effects. equationsof aberrationcoefficients, it yields the expressionsof aberrationsdepending on the 5. Conclusions bending factor. Therefore, the best way to This paperdiscusses the use of Lie Methods removethe aberrationsis by bendinglenses and to characterizethe aberration of third order of by using a system of compound lenses. For the macrolens in stage II of the OTIS. This example,with two thin lensesin contact,it is approach, like that of Hamiltonian optics, possible to eliminate both sphericalaberration embodiesthe restrictionsimposed by Fermat's and coma,by judicious choiceof the lens shape. principle. Moreover, it has the feature that the The macrolensin stageII is a bi-convex single relation between initial and final quantitiesis lens.By usingthese results the macrolenscan be alwaysgiven in explicit form. In particular,it is designedusing CODE V (lens design software, possibleto representthe action of each separate Optical Research Associates, U.S.A.) as a element of a compound optical system, compoundlens systemwith 6 lensesas shownin including all departuresfrom Gaussianoptics, Figure4. by a certain operator. These operators can then Figure 4 showsa Fourier transformlens [8] be concatenatedby well-defined rules of Lie which performs the same function (Fourier algebraicmethods to obtain a resultantoperator transform)as the bi-convex single lens in stage that characterizesthe entire system. That is, the II does. This compound lens system can use of Lie methods provides an operator eliminate almost all aberrationsas shown in extension of the matrix methods of Gaussian Table 2 below. opticsto the generalcase. Finally, their use,as is the casewith characteristicfunctions, facilitates the treatmentof symmetriesin an optical system and the classificationof aberrations.This is one of the major advantagesof Lie methods to provide additional insight concerning the sourcesof various aberrations.The expressions for the primary (third order) aberrations of a single lens dependon the surfacecurvatures of the lens. The dependenceof abenationson the surface curvaturescan be expressedin terms of the shapefactor of a thin lens. The shapefactor of a thin lensis definedas

'l (30) Figure 4. A Fourier Transform Compound X ='L ct-cz Lens. ThamgrasatInt. J. Sc. Tech., Vol. 9, No. 2, April-June 2004

Fourier TraEform Lans Ver I Posj,Cion 1. gavelengEh = 532.8 Ntr 5l' TCO TIS SIS PTB DsT l'X Ll'T PTZ sTo o.000000 0. 000000 0. oo0000 0.000000 0.000000 o.ciooooo o,oooooo o,000000 0. ooo000 2 O.L32lO9 -O.32,1151 O,415215 O.239469 O.151095 -0. 195A59 0. OBOOOO o. 000000 0. 001315 3 4.35?s6? 6.165562 3,{02359 L.259As3 0.203600 o,6281{8 0.000000 o.000000 0,ooL7'72 1 -L.276026 -2,A23565 -2.L35916 -O.?4?5L7 -0.053303 -o. 551354 0. 000000 o. 000000 -0, 000464 5 -0.04413{ 0.110863 -O.209266 -0.1?5803 -0.1505?2 o.01?6s7 0, oooooo 0, 000000 -0. o0139? 6 -O.?15343 -1.698?04 -1.5391?3 -0.542809 -O.L94621 -0.508?92 U. OOOOOO o,000000 -0.0o1693 "t -o,265263 0.,1O3,186 -O,206"12't -O.010312 -O.OO21{9 o,035665 0.000000 o.000000 -o.000019 a 0,15?886 -0,2?0596 0.L43232 0.O,16311 -0.0O2149 -0.024881 0.000000 o . ooooo0 -o . 0000 19 9 -2.1s3162 -?.A3L?1-A -1,16s360 -0.518205 -O.79162? -o, \71544 0.000000 o . oooo00 -0 . oo 1593 10 0.210645 -O.391930 0.O82505 -O.O?9545 -O.L605't2 o. 049335 0. OOOOOO o,000000 -0. o0139? 11 -1.549839 -1.?10493 -0,6825?0 -0.253059 -0.053303 -0,096??5 0.000000 o. oooooo -o,00046{ L2 1.?35852 2.856966 1.??B9A? O.?26063 0.203600 0.398332 0. UOOOO0 0.000000 0.oot7?2 13 -O.151?O9 0.15161{ O.100589 0,134260 0.151095 lo,011?25 0.OOOOO0 0.000000 0,o01315

suu -0,191?51 -O.06166? -O,O?3154 -O.082325 -0.111910 -o.410904 0. ooo000 o. ooo000 -o. ooo9?4

Table 2. Third order Aberrations of the compound lens

6. References [7] The pictures in Figure 2 are simulated by LightTools which is an illumination design [1] Kurt Bernardo Wolf, Symmetry in Lie program based on 3D interactive solid Optics,Annals of Physics,Yol. 172, pp. l- modelling with optical 25,1996. accuracy. The software is developed by Optical Research [2] Tetsundo Sekiguchi and Kurt Bernardo Associates,www.opticalres.com. Wolf, The Hamiltonian formulation of W. Pijitrojana, Optical Transpose optics, Am. J. Phys.,Vol. 55, pp. 830-835, t8l Interconnection System Design 1987. in Optical Phase Space, A Ph.D. thesis submitted to [3] Octavio Castanos,Enrique Lopez-Moreno, the Universityof London,2002. and Kurt Bernardo Woll Canonical t9l Alex J. Dragt, Lectures on Nonlinear Orbit transforms for paraxial wave optics, Lie Dynamics,University of Maryland,College Methods in Optics, Proceedings,Leon, Park, Maryland, American Instifute Mexico, edited by J. SanchezMondragon of Physics,1982. and Kurt Bemardo Wolf, Lecture Notes in Stanly Steinberg, Lie Physics,Springer-Verlag, 1985. [0] series, Lie transformations,and their applications, Lie t4l W. Pijitrojana and T. J. Hall, Optical Methods in Optics, Proceedings,Leon, Transpose Interconnection System Mexico, edited by J. Sanchez Mondragon Architecture, The 26- Electrical and Kurt Bernardo Engineering Conference (EECON-26), Wolf. Lecture Notes in Physics,Springer-Verlag, 1 985. Bangkok, Thailand, Vol. III, pp. 1599- 1604.2003. [ll]Alex J. Dragt, Etienne Forest, and Kurt Bernardo Wolf, Foundations of a Lie [5] A Gerrard and J. M. Burch, Introduction to algebraic theory of geometrical optics,Lie Matrix Methods in Optics, Dover Methods in Optics, Proceedings, Publications,Inc., 1975. Leon, Mexico, edited by J. SanchezMondragon [6] Kurt Bernardo Wolf, Fourier transform in and Kurt Bernardo Wolf. Lecture metaxial geometric optics, J. Opt. Soc. Am. Notes in Physics, A, Vol. 8, pp. 1399-1403,1991. Springer-Verlag,I 985.

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