Design Alternatives for a Thin Lens Spatial Integrator Array

Transcription

1 Egyp. J. Solids, Vol. (7), No. (), (004) 75 Design Alernaives for a Thin Lens Spaial Inegraor Array Hala Kamal *, Daniel V azquez and Javier Alda and E. Bernabeu Opics Deparmen. Universiy Compluense of Madrid. School of Opics. Av. Arcos de Jal on s/n Madrid. Spain. In his presen paper, we presen a heoreical analysis relaing he ype, power and he maximum number of he idenical elemens in he inpu surface of he array, wih is size, flux ransfer efficiency ono he synheic image plane and he synheic image qualiy. The numerical analysis resuls enabled he design of wo differen planar hin-lens spaial inegraor arrays wih opimized geomerical parameers and wih differen posiive and negaive power values. A compuer program has been developed o invesigae he performance of each array. The percenage of he ougoing rays, he average ransmiance of he array elemens, he image widh and he uniformiy of he ransferred energy, have been evaluaed and compared. - Inroducion: Lens arrays, formed eiher by refracive or diffracive lenses, are increasing in imporance because hey have a lo of new uses in modern aspecs of opics. They have aroused considerable ineres in he area of conrolling beams by allowing individual elemens o have heir own focusing and direcing elemens. Compared wih a convenional opical sysem, an opical array has he advanage of working in parallel; i divides he inciden wave fron ino porions which proceed hrough he individual opical elemens o form a composed image. This composed image is called synheic image and generally i doesn coincide wih he image of he objec given by he individual elemens. This sudy is focused on an ineresing ype of opical arrays called spaial inegraors. * On leave From Physics Deparmen, Faculy of Science, Ain Shams Universiy, Cairo, Egyp.

2 Hala Kamal. e al 76 They are non-imaging devices wih zero ray ransfer marix deerminan, so hey are called arrays wih vanishing deerminan. The essenial characerisics of such arrays are, having infinie field deph and an image size independen of he posiion of he objec. In oher words, all sources in he objec space are imaged a he same plane. Despie of being non imaging devices, hey found oher several applicaions. They are used o ransfer radiaion from an exended source o a arge in such a way as o achieve a specified disribuion of radiaion on he arge. They are employed as energy concenraors in solar furnaces and as illuminaors in shadow-less lamps. Wihin he marix opics frame work [, ], he firs design condiion for a ligh inegraor array is ha he disance R beween he inpu plane and oupu plane of he array should obeys he following formula R b l d = () where l is he lengh of he individual opical elemen, and b and d are he elemens of he marix of each individual elemen of he array. In previous research, [3, 4, 7], we have developed several ideas o improve he efficiency for hese kinds of arrays. For a spherical hick lens spaial inegraor arrays he design parameers are opimized in such a way o ransfer as much as possible he energy inciden on he array inpu plane o he synheic image plane. Also, a planer arrangemen was proposed [5,.6, 7] which is easier o be fabricaed and inegraed ino he opical sysems. The purpose of his paper is o explore he influence of he ype, power of he array opical elemens and he maximum number of array elemens on he inpu surface of he array, on he ransferred flux and he image qualiy formed ono he synheic image plane (S.I.P.). Planer hin-lens ligh inegraor arrays are sudied. The firs secion deals wih a deailed heoreical analysis which explore he influence of he ype of he opical elemen (diverging or converging) in he inpu surface of he array, is power and he maximum number of idenical elemens of he array ha can work ogeher, in order o ransfer he inciden flux ono he S.I.P., regardless he individual efficiency of each array elemen. Secondly, a numerical analysis was done based on he heoreical relaions, obained in he previous secion, giving rise o differen resuls describing, boh he behavior of a planer hin-lens ligh inegraor array and he behavior of he individual elemen a differen power values. Thirdly, he heoreical predicions as well as he numerical resuls obained previously, allows he design of wo differen hin-lens planar ligh inegraor arrays. Performance as well as comparison of some major aspecs of he wo designed arrays are given in he fourh secion.

3 Egyp. J. Solids, Vol. (7), No. (), (004) 77 - Planar Thin lens Spaial Inegraors: A scheme of a planer hin lens spaial inegraor array is shown in Fig.(). I is composed of a pair of planar hin lens arrays (PTLA) separaed by a disance ; he pich of he inpu plane (PTLA) is longer han ha of he oupu plane (PTLA). The individual elemen of he array is composed of a couple of hin lenses of differen focal lenghs f and f. The opical axis of each elemen is he line joining he ceners of he opical elemens forming he uni. Each elemen has is own opical axis, all inersecs a he cener of he synheic image plane. The ype of he elemens on PTLA could be eiher converging, divergen or boh lenses PTLA k = 3 PTLA k = k = Array General Axis k = - Elemen Axis k = - k = -3 R Fig. (): A ligh inegraor array composed of a couple of planar hin lens arrays (PTLA &PTLA). I follows from equaion () ha, for a hin-lens ligh inegraor array, he disance beween he inpu plane and he synheic image plane of he array is, f R = () f

4 Hala Kamal. e al 78 The las equaion indicaes ha he spaial inegraion feaure of his ype of arrays doesn depend on he focal lengh of he firs hin lens f which is lef as a free parameer in he design of he array. Insead, he value of f and should be fixed. Very recenly [8], he focal lengh values for PTLA opical elemens are opimized in such a way o maximize he ransfer of ligh energy inciden upon i o he synheic image plane (S.I.P.). Anoher crierion ha migh be aken ino accoun, is he maximum number of elemens ha can work ogeher in order o ransfer ligh-rays o he synheic image plane for a given value of f. In his case, we assume ha all he elemens of PTLA have equal focal lengh f and we ake ino consideraion all elemens ha permi he ransfer of any par of ligh-rays inciden upon i o he synheic image plane. For he planer hin lens-array configuraion given in Fig. (), as he axis of he individual elemens become more and more misaligned, he efficiency of he elemens o ransfer radiaion o he synheic image plane becom smaller. In he following we obain he maximum order of he elemens ha a planer hin-lens spaial inegraor array can bear, wo differen cases are considered. Case () : Posiive planer inpu array: In his case we assume ha, PTLA is composed of converging hin lenses of equal focal lengh f and he elemens of PTLA are of idenical focal lengh f, in order o comply wih he vanishing deerminan condiion of equaion (). PTLA and PTLA are assumed o be of differen pich and respecively, and >. For posiive PTLA wih f values greaer han he individual elemen size. The maximum order ha he individual elemen can aain is defined by h. he ray inciden a he lower exreme of he k elemen of PTLA and afer refracion goes hrough he upper exreme of he elemen in PTLA, Fig. ()a. Considering a parallel bundle of rays aligned wih he array general axis, his can be expressed in a maricial form as; where, ω k = f 0 0 (3)

5 Egyp. J. Solids, Vol. (7), No. (), (004) 79 The maximum order of he elemen for ( ) k = ω k (4) f > ; = Pr eviousineger f ( f ) ( ) k (5) Similarly, for posiive PTLA wih f values smaller han he individual elemen size. The maximum order ha he individual elemen can be calculaed by considering he ligh ray inciden a he maximum heigh of he h. k elemen of PTLA o be deviaed wih an angle θ k, Fig. (-b). Considering a parallel bundle of rays aligned wih he array general axis, his can be wrien in a maricial form as; where, θ k = f ( ) 0 0 (6) k θ k = (7) The maximum order of he elemen for f < ; k = Pr eviousineger (8) f( ) Case () : Negaive planer inpu array For a negaive planer spaial inegraor array, PTLA is assumed o be composed of diverging hin lenses of focal lengh f, all sacked side by side in a planer arrangemen and separaed from PTLA by a disance. For >, he ligh-ray ha defines he maximum number of elemens of he array is ha which ouches he h. k elemen of PTLA a and hen

6 Hala Kamal. e al 80 deviaes by an angle ϕ k where i passes hrough he corresponding elemen on PTLA, Fig.()c. Considering a parallel bundle of rays aligned wih he array general axis we ge; Figure ( a) Figure ( b) PTLA PTLA f f ω k θ k PTLA PTLA PTLA -f α k PTLA Fig. (): Scheme of a ligh inegraor array elemen composed of a couple of hinlenses, he oupu lens has a posiive focal lengh f. The inpu lens has, (a) Posiive focal lengh f >. (b) Posiive focal lengh f <. (c) Negaive focal lengh. α k = f 0 0 (9) where; ( ) k α k = (0)

7 Egyp. J. Solids, Vol. (7), No. (), (004) 8 The maximum order of he elemen is given by; = Pr ( + f) ( ) k eviousineger () f Noe ha he maximum number of elemens of he array is wice he maximum order of he elemen k, as he elemens are arranged up and down he array general axis. 3. Simulaion and Resuls: In order o explain and clarify he imporance of he Previously obained relaions, we have invesigaed he behavior of hin lens spaial inegraor arrays wih maximum number of elemens wih differen posiive and negaive PTLA power values. The inpu planar hin lens array (PTLA) is assumed o be of idenical elemens. A compuer program has been developed o independenly esimae he behavior of each array. The geomerical parameers of he array used in he compuer simulaion are given in Table (), The values have been chosen according o ligh inegraor arrays opimizaion heory presened recenly in [8]. In Fig. (3), we have ploed he power of PTLA in diopers (D) versus: (a)the maximum order of he elemen obained from he above formulas, (b) The percenage of he ougoing rays developed from each array wih respec o he incoming rays, (c) The average ransmiance of all array elemens,(d) The image widh (ϖ ) and he coefficien of uniformiy (u) for he energy disribuion on he synheic image plane φ (x), defined as u = paraxial φ( x) dx ( x) () Noe ha, he inegraion is aken only on he spaial region of he paraxial synheic image plane. This parameer equals zero for a perfec uniform disribuion wihin he paraxial synheic image. I is clear from he resuls depiced in Fig.(3)d ha, alhough he uniformiy of he image varies significanly wih he power of he individual elemens in PTLA, he image size obained from he numerical simulaion changes slighly which coincides wih he heoreical predicions of he paraxial size of he synheic image, repored previously [], = (3) f

8 Hala Kamal. e al 8 Table (): Geomerical parameers of a planer hin-lens ligh inegraor array used in he compuer simulaion. The refracive index of he lenses maerial is.53. R 80 mm f 37.9 mm 80 mm 70 mm 60 mm 8 Maximum Order Of The Elemen Power Of PTLA Individual Elemen 90 Percenage Of Ougoing Rays Power Of PTLA Individual Elemen

9 Egyp. J. Solids, Vol. (7), No. (), (004) Average Transmiance Of The Array Power Of PTLA Individual Elemen Coeff. of Uniformiy Normalized Image Widh Coeff. of Uniformiy Normalized Image Widh Power Of PTLA Individual Elemen Fig. (3): Relaion beween he power of PTLA elemens Versus: (a)the maximum order of he elemen. (b) The percenage of he ougoing rays. (c) The average ransmiance of he array. (d) The image widh (ϖ ) and he coefficien of uniformiy (u).

10 Hala Kamal. e al 84 Also, he behavior of each array elemen a differen focal lengh values has been found whereas he power ( F ) of PTLA have aken differen posiive and negaive values, namely; ±, ±, ± 3,..±5 diopers (D). As a measure of efficiency of each individual elemen, he percenage of he ougoing rays wih respec o he incoming ones as well as he average ransmission of he array have been calculaed for differen power values and depiced in Fig. (3) b and c. Boh curves show peaks a power range 5-7 D for he PTLA individual elemens. For clariy, he number of he ougoing ligh-rays wih respec o he inciden rays has been compued a differen order of he elemen and represened in Fig.(4). I shows ha all curves converge a a paricular order of he elemen hen diverge again. The advanages of he previously obained graphs is ha, i enables he design of differen spaial inegraor arrays adequae for differen purposes. In oher words, hese graphs can be used as a guide in arrays design. In he following, wih he aid of he obained resuls, design of wo differen spaial inegraor arrays is given. Alhough, in he previous graphs he power of hin lenses in PTLA has aained large values, in designing he following wo arrays we have considered hin lenses in PTLA wih sop number given by; F number (4) = f smaller han wo as hey are easier o be fabricaed). 4. Design: Ligh inegraor arrays have several applicaions leading o differen kinds of arrays. Alhough, each kind depends on he deails of he opical sysem, some crierion could be laid down which enable defining he global behavior of he array. Those crierian comprise; (a) Number of array elemens (size of he array). (b) Type of elemens on he array inpu surface (for spaial inegraor arrays he opical elemens of he inpu surface doesn' affec he inegraion feaure). (c) Efficiency of each elemen. (d) The uniformiy of he image on he synheic image plane and formed by all array elemens. (e) The uniformiy of he image produced by all array elemens.

11 Egyp. J. Solids, Vol. (7), No. (), (004) 85 Percenage Of Ougoing Rays D D 4D 6D 8D 0D D 4D 6D 8D Percenage Of Ougoing Rays Order Of The Elemen -D -D -4D -6D -8D -0D -D Order Of The Elemen Fig. (4): The order of each array elemen in PTLA versus he percenage of ougoing rays; (a) For posiive PTLA power values. (b): For negaive PTLA power values.

12 Hala Kamal. e al 86 Considering hose crierion and wih he aid of he obained resuls represened in he above figures, wo differen planer hin-lens ligh inegraor arrays are designed. i- The firs spaial inegraor array has been designed wih maximum number of idenical elemens on PTLA offering maximum: ransmiance, number of ougoing rays wih respec o incoming rays, image widh, image uniformiy and energy on he synheic image plane (see plo (3)). For his array, PTLA has eigh opical elemens (up and down he array opical axis) each wih a power equal o five diopers, ii- For he second array, he elemens in PTLA are no idenical. I is designed o ransfer as much as possible energy ono he synheic image plane regardless is uniformiy. For each opical elemen in PTLA, we have considered is individual behavior a a given posiion (order) wih a definie power. Referring o Fig. (4), he hin-lens elemens in PTLA were chosen o have he powers given in Table () corresponding o each elemen order. The maximum order of he elemen in his case is eleven. As he opical elemens are arranged on boh sides of he array general axis, he oal number of elemens for his array is weny wo elemens. Three facors were aken ino accoun in designing such array. Firs, increasing he maximum number of elemens (array area) ha can work ogeher o produce he synheic image. Second, obaining maximum efficiency. Finally, choosing (as much as possible) long focal lengh elemens on PTLA as i offers less spherical aberraions and reduced asigmaism. Table (): Power of he hin-lens elemens in PTLA corresponding o each order. Case (a): array wih maximum order of idenical elemens. Case (b): array wih unidenical elemens. Order of he elemen Power of lenses on PTLA. case (a) Power of lenses on PTLA.case (b) In order o compare he behavior of he wo previously designed ligh inegraor arrays, a developed compuer program has been used o calculae he energy disribuion ono he synheic image plane, normalized o maximum value, and hen presened in Fig. (5), he posiion is being normalized o he paraxial size of he image. The coefficien of uniformiy (wihin he image paraxial size), corresponding o each energy disribuion, he percenage of he ougoing ligh-rays and he energy delivered ono he synheic image plane have been calculaed and presened in Table (3).

13 Egyp. J. Solids, Vol. (7), No. (), (004) PTLAof Idenical Elemens PTLAof Unidenical Elemens Normalized Energy Normalized Posiion On The S.I.P. Fig. (5): Energy disribuion ono he S.I.P. normalized o he maximum value for wo planar hin-lens spaial inegraor arrays wih idenical and unidenical elemens of PTLA. Table (3): Coefficien of uniformiy of he energy disribuion ono he S.I.P. and he percenage of ougoing rays of planar hin lens ligh inegraor arrays wih (a) idenical, (b) unidenical elemens. Geomerical parameers are given in able (), powers of PTLA elemens are given in Table (). Case Coeff. of uniformiy u Energy on he S.I.P (Arb.U.) Percenage of ougoing rays a b Conclusions: By sudying he behavior of ligh inegraor arrays formed by a couple of planar hin lens arrays, we have found how he ype and power of he inpu array surface influence is; (a) size (maximum number of elemens), (b) Efficiency of he array represened by he amoun of energy and he percenage of ougoing rays wih respec o incoming rays delivered ono he synheic

14 Hala Kamal. e al 88 image plane, (c) Synheic image qualiy evaluaed by image widh and coefficien of uniformiy. The significance of he previously obained resuls is ha i enables he design of wo differen examples of planer hin lens ligh inegraor arrays adequae for differen array applicaions. Acknowledgmen This work has been done duran he say of Dr. Kamal in Opics Deparmen, Universidad Compluense De Madrid. The auhor would like o hank Dr. Daniel Vazquez for his suggesions and commens abou his paper. References:. Wang Shaomin, Laura Ronchi, Progress in opics, 5, Norh-Holand, Amserdam (988) 79.. S. Wang, D. Zhao, Marix Opics, Springer-Verlag, Heidelberg, (000). 3. JavierAlda, HalaKamal, Eusebio Bernabeu, Op. Eng. 36, (997) Hala Kamal, design and properies of opical arrays, PhD disseraion, Universidad Compluense de Madrid, Spain, (998). 5. Daniel Vázquez, Eusebio Bernabeu, Lig.Res.Tec, 9 (), (997) Daniel Vázquez, Javier Alda, and Eusebio Bernabeu, Applied Opics 38, (999) H. Kamal, Opik 4, (003) Hala Kamal, Daniel Vázquez and Javier Alda, Egyp. J. Solids, Vol. 7 (), (004) 35.