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