Janus Waves. Department of Material Science and Technology, University of Crete, P.O. Box 2208, 71003, Heraklion, Greece 3

Transcription

1 Janus Waves DIMITRIS G. PAPAZOGLOU, 1, VLADIMIR YU. FEDOROV, 3,4 STELIOS TZORTZAKIS 1,,3 1 Institute f Electrnic Structure and Laser, Fundatin fr Research and Technlgy-Hellas, P.O. Bx 157, 71110, Heraklin, Greece Department f Material Science and Technlgy, University f Crete, P.O. Bx 08, 71003, Heraklin, Greece 3 Science Prgram, Texas A&M University at Qatar, P.O. Bx 3874, Dha, Qatar 4 P.N. Lebedev Physical Institute f the Russian Academy f Sciences, 53 Leninskiy Prspekt, , Mscw, Russia dpapa@materials.uc.gr We shw the existence f a family f waves that share a cmmn interesting prperty affecting the way they prpagate and fcus. These waves are a superpsitin f twin waves, which are cnjugate t each ther under inversin f the prpagatin directin. In analgy t hlgraphy, these twin real and virtual waves are related respectively t the cnverging and the diverging part f the beam and can be clearly visualied in real space at tw distinct fci under the actin f a fcusing lens. Analytic frmulas fr the intensity distributin after fcusing are derived, while numerical and experimental demnstratins are given fr sme f the mst interesting members f this family, the accelerating Airy and ring-airy beams The frmatin f twin images, a real and a virtual ne, as light is diffracted by a hlgram is a well knw effect [1]. The hlgram acts as a spatial mdulatr f the amplitude and phase f the riginal wavefrnt, and thus a cmplex wavefrnt is frmed. This wavefrnt can be decmpsed int tw waves [], ne f which leads t the frmatin f the real image f the riginal bject at sme distance and the ther t the frmatin f the virtual image at distance -. These tw inseparable waves interfere, and may deterirate the recnstructed bject. In hlgraphy the twin image effect is undesirable because f the deteriratin f the images and has been successfully bviated [3] by intrducing a separate tilted reference wave during the hlgram recrding prcess. Here we shw that the frmatin f symmetric twin images can be bserved nt nly in hlgrams but als fr a brad class f waves that we will call in the fllwing as Janus Waves (JWs). Like the gd Janus frm the Rman mythlgy, wh was depicted with tw faces lking in ppsite directins JWs can be decmpsed t the prpagatin f tw waves, which are cnjugate t each ther under inversin f the prpagatin directin. Such waves include beams like the accelerating Airy [4-6] and ring-airy beams [7-11] and higher rder accelerating beams [1-16]. A frmal definitin f JWs will be prvided in the fllwing. The analgy t hlgraphy hlds if we cnsider the intensity peaks, r fcus pints, f these waves as an bject. In this case the frmatin f twin images manifests as tw fci a real and a virtual ne symmetrically psitined t the plane f symmetry. Interestingly this effect was, t ur knwledge, nt bserved up t nw. In cntrast t classical hlgrams, and due t the gemetry f the beam, the part f the wave that is related t the virtual image is diffracting ut with minr interference with the part f the wave that is related t the real image. The presence f the twin virtual image clearly manifests when these beams are fcused by a lens. In this case bth the cnverging and the diverging part f the beam are brught t fcus and frm a pair f ppsite facing fcal distributins. Theretical analysis Let s assume that a harmnic wave described by its field U( r, ), where r is the transverse psitin vectr and is the psitin alng the prpagatin axis, can be described as the superpsitin f tw waves, ( r, ) and a cnjugate ( r, ) : U ( r,) ( r, ) ( r, ) (1)

2 These cnjugate waves (CW) are in perfect analgy with in-line hlgrams, where the diffracted wave is frmed by the superpsitin f the real and virtual image. One can easily shw that this wave is cnjugate symmetric under inversin f the prpagatin directin, U( r, ) U ( r, ). Furthermre fr such a field the distributin at =0, which defines the plane f symmetry, is always a real valued functin: U ( r,0) ( r,0) ( r,0) Re[ ( r,0)] The mathematical criterin fr CWs, is simple and clear and is based n eq. (): A wave is Cnjugate if its field U( r, ), is real valued at a transverse plane alng its prpagatin. By real valued we mean that the phase can take nly m values (where m 0, 1, ). In what fllws we prve the validity f the CW criterin. Withut lss f generality we can set the rigin f ur crdinate system n a plane s that: U( r, 0) Re{ U( r,0)}. The prpagatin f such a wavefrnt can be described using the cncept f the angular spectrum []. The angular spectrum i( kxxkyy) P ( k ) U ( x, y,0) e dxdy, where k is the transverse cmpnent f the wave vectr, f such a real valued field is Hermitian P described as: i P ( ) P ( ) e k k ( k ) ( ) P( ) () k k and thus can be where the amplitude P( k ) P( k ) f the angular spectrum is an even functin, and ( k ) ( k ) is an dd phase distributin in the k-space. Using the angular spectrum at 0 we can estimate the amplitude f the wave at any distance alng the prpagatin axis: i ( ) i( kxx kyy) k i k k k x y U ( r, ) P ( ) e e e dk dk ( r, ) ( r, ) 1 i ( ) ( ) k i kxx kyy i k k where ( r, ) P ( ) e e e dkxdk y k. Thus the prpagatin f a wave that is real valued, at a transverse plane alng the prpagatin, can always be described as the superpsitin f tw waves, ( r, ) and a cnjugate ( r, ). A large variety f waves fulfill the CW criterin, ranging frm accelerating beams such as Airy [4, 17] and ring Airy beams [7, 8], but als the widely used Gaussian beams and Bessel beams [18]. On the ther hand Helical beams [19], are waves that d nt fulfill this criterin due t their helical phase gradient. Frm this brad categry f CWs we nw cncentrate n thse waves that exhibit a discrete fcus away frm their symmetry plane. We define such waves as JWs. These include fr example all accelerating beams [4-9, 1, 13] but nt Gaussian and Bessel beams [18]. What makes JWs interesting is their peculiar behavir when they are fcused by a lens. As we will shw in the next sessin, tw fci, instead f ne, can be bserved. A characteristic example f a JW is the ring-airy beam [7]. These beams fulfill the CW criterin and exhibit a discrete fcus. Mre specifically, they are real valued at their generatin plane since they are described [7, 8] by Ai( ) e a, where Ai(.) is the Airy functin, ( r r ) / w and r is the radius, r, and w are the ring radius and width parameters respectively and is the apdiatin cefficient. Furthermre, they als exhibit a discrete fcus, away frm their symmetry plane, since they abruptly autfcus [7, 8] at predefined distance [9]. (3) (4)

3 Fig. 1. Prpagatin f Janus waves as a superpsitin f real and virtual waves (a) graphical example f a JW under the frm f a ring-airy beam. (the tw-faced Rman mythlgy gd Janus is als represented) (b), (c) Crss sectinal (r-) simulatin results f ring-airy prpagatin. The parablic trajectry twards the fcus is clearly visible bth in the intensity (lgarithmic scale) and the phase (wrapped phase). The virtual wave existence althugh practically invisible in the intensity map is clearly visible in the phase A graphical example f a JW prpagatin under the frm f a ring-airy beam is shwn in Fig. 1(a). The fcal pint lies at a distance while a virtual fcal pint lies n the ppsite side (at - ) and is nt accessible. The effect f the twin real and virtual waves n the beam prpagatin can be visualied by numerically simulating the prpagatin f a ring-airy beam. As shwn in Figs. 1(b),(c), the characteristic parablic trajectry twards the real fcus is clearly visible bth in the intensity (Fig. 1(b)), and the phase f the beam (Fig. 1(c)) as depicted by, the phase wrap uncvered, is-phase curves. On the ther hand, a signature f the existence f the virtual wave can be seen nly in the phase map. The intensity, althugh in lgarithmic scale, shws n trace f the cmpnent that diffracts ut. The virtual wave manifests as a diffracting ut wave and is usually ignred [16]. If we culd reverse prpagate this diffracting part then we wuld bserve a symmetric t the initial fcus, abrupt fcus. Janus wave fcusing by a lens If we insert a thin lens n the plane f symmetry ( 0) f a JW then the wave distributin after the lens is described by [0]: ik ( x y ) 1 f r r, e,, Ur where 1 / f. The abve was calculated using the ABCD matrix descriptin f a thin lens f fcal distance f. The intensity distributin after the lens can then be estimated by: 1 r r r r I,,, Re,, r (6) (5)

4 Fig. Fcusing a Janus wave with a thin lens. Withut the presence f the lens each intensity peak (fcus) is accmpanied by a symmetrically psitined virtual intensity peak. The fcusing actin f the lens creates tw ppsite facing fcal distributins. As expected the resulting intensity distributin is the result f the interference f the tw waves. Interestingly, a lcal maximum f the wave distributin ( r, ) at psitin ( r, ) will be imaged after the lens at tw psitins ( r 1, 1 ), ( r, ):, r , r r r 1 1 1, r1 M ( ) r 1 f 1 1 1, r MT ( ) r f T 1 where i 1 i / f, and ( ) (1 / ) MT f is the transverse magnificatin f the lens. Althugh the accurate lcatin f the intensity maxima frm eq. (6) invlves a cmplex analysis, the psitins f the twin images, as given by eq. (7), are crrect as lng as 0, 1 / f 0.1. Cunterintuitively, classical imaging frmulas can i be used t predict the maxima psitins if ne takes int accunt that the peak f the ( r, ) wave can be treated as an imaginary bject t the lens ( as a real bject ( ). ) while the peak f the cnjugate ( r, ) wave can be treated Furthermre, the transverse psitin f the peaks is predicted by taking int accunt the transverse magnificatin M T fr each psitin f the bject. It is nt difficult t see that the tw fci will exhibit ppsite signs f M T fr f. As shwn schematically in Fig. withut the presence f the lens each initial intensity peak (fcus) f the wave is accmpanied by a symmetrically psitined virtual fcus. The fcusing actin f the lens leads t the creatin f tw ppsite facing fcal distributins lcated at psitins 1, that can be estimated using eq. (7). The tw fci are distinct as lng as the twin waves, ( r, ) and ( r, ) d nt have verlapping maxima alng. It is easy t shw that verlapping maxima can ccur nly n the symmetry plane 0, a case already excluded frm the definitin f JWs. Thus a JW, in cntrast t the brader categry f CWs, will always exhibit tw fci under the actin f a lens. The validity f ur analysis can be cnfirmed by applying it in the case f clliding 1D Airy beams[1]. These beams are the 1D equivalent f D rtatinally symmetric ring-airy beams [7, 8, ] and are cnsisting f tw 1D Airy beams that prpagate alng symmetric parablic trajectries. (7)

5 Fig. 3 Intensity distributin f 1D clliding Airy beams. Cmparisn f analytical predictin with numerical simulatin results (a) Free prpagatin, (b), (c) prpagatin after being fcused by a thin lens. (parameters x =1 mm, w = 84.5 m, = 0.05, = 800 nm, f = 00 mm) The tw beams cherently interfere generating a fcal pint n the prpagatin axis. In this case the beam prpagatin can be analytically described [4] by: U ( x, ) ( x, ) ( x, ), 1D x x a ( x, ) Ai( i )e w 4k w k w 3 xx a a xx a i( ) w k w k w kw 1k w 4 4 where x is the spatial crdinate, x and w are respectively the primary lbe psitin and width parameters and k is the wavenumber. It is clear that at 0 the wave is real valued. In this case, fllwing eq. (1), the wave can be decmpsed t the interference f tw waves, ( x, ) and a cnjugate ( x, ) by simply setting: ( x, ) U ( x, ) /. 1D (8) As shwn in Fig. 3(a) the symmetric Airy beams prpagate fllwing a parablic trajectry generating a fcus at 400 mm. When this beam is fcused by a lens, the resulting intensity distributin is a duble fci image tgether with an interference pattern between the, cnjugate waves. Cmbining nw Eqs. (8) and (5), adapted fr 1D prpagatin, we can analytically frmulate the wave distributin after the insertin f a thin lens at 0. The analytically estimated intensity distributin (frm eqs. (8), (5)) is shwn tgether with numerical simulatin results in Figs. 3(b),(c), and as ne can clearly see the agreement is excellent. Furthermre, the tw fci are lcated at 1 = 133 mm and = 400 mm as predicted by eq. (7). Experimental results Fr the experimental demnstratin f JWs we have chsen t wrk with ring- Airy beams [7]. The experimental setup used t study this effect is shwn in Fig. 4. A cw laser dide emitting at 800 nm was used as a surce. In rder t generate the ring-airy beam we fllwed a Furier Transfrm apprach described in detail in [8]. In brief, we first used a phase nly reflecting spatial light mdulatr (SLM) in rder t mdulate the phase f a Gaussian beam. The mdulated beam was then Furier Transfrmed by a lens. The ring-airy distributin is generated in the Furier transfrm plane f the lens after blcking the er rder diffractin. We have tuned the parameters f the ring-airy beam s that the abrupt autfcus psitin was at =400 mm.

6 Fig. 4 Experimental setup. FT: Furier Transfrm. FT lens (500 mm). fcusing lens f: 00 mm, (the beam prpagates frm left t right) The transverse intensity distributin as the beam prpagated was imaged using a cmpact micrscpe system (0. NA) and a linear CCD camera (14 bit). We firstly studied the ring-airy beam prpagatin withut the presence f a fcusing lens and then we fcused this beam by placing a plan-cnvex lens f fcal distance f = 00 mm clse t the ring Airy generatin plane. The I(x,y,) intensity prfile f the beam was retrieved by merging D transverse (x,y) images captured at varius psitins alng the prpagatin axis (). An x- crss sectin f the intensity prfile is shwn in Fig. 5 tgether with numerical simulatins f the paraxial wave equatin. As expected frm ur frmer analysis fr JW waves the fcused ring-airy beams frm tw fcal regins. The real and virtual waves prduce tw fci at different psitins and the psitins f the tw fci agree well, within %, t the theretical predictins f eq. (7). Finally, as can be seen in Fig. 5 the agreement between the experimental measurements and the simulatins is remarkable. Fig. 5 Crss sectinal intensity prfiles I(x,0,) f ring-airy beams. Experimental measurements (tp rw), simulatin (bttm rw). The simulatins are perfrmed assuming a thin lens lcated at psitin =0. Free prpagatin (left clumn), fcusing by a 00 mm lens (right clumn) Cnclusins We have intrduced a family f waves, called Janus waves, whse prpagatin can be decmpsed t real and virtual cmpnents, cnjugate t each ther under inversin f the prpagatin directin. Frm the theretical pint f view, we have established simple criteria fr a wave t belng t the Janus wave family. On the ther hand, frm the experimental pint f view Janus waves can be easily identified by the appearance f tw fcal regins after the actin f a fcusing lens.

7 We have derived analytic frmulas fr the intensity distributin after the actin f a fcusing lens, while interestingly, simple imaging frmulas can be used t predict the intensity peak psitins. Finally, this exciting behavir is demnstrated bth experimentally and by simulatins fr the case accelerating ring-airy beams. Due t the equivalence t hlgraphy, ur apprach can be generalied t all waves that rely n diffractin t generate an intense fcal pint. We expect that Janus waves will find exciting applicatins bth in linear and nnlinear ptics. The pssibility t engineer symmetric fci distributins can fr instance impact ptical trapping applicatins, r the cntrlled depsitin f high laser pwers at remte lcatins. Funding. QNRF prject N. NPRP , Laserlab-Eurpe (EU-FP ) References 1. D. Gabr, Nature 161, 777 (1948).. J. W. Gdman, Intrductin t Furier ptics, 3rd ed. (Rberts & Cmpany, 005). 3. E. N. Leith and J. Upatnieks, JOSA 5, 113 (196). 4. G. A. Sivilglu, J. Brky, A. Dgariu, and D. N. Christdulides, Phys. Rev. Lett. 99, (007). 5. D. Abdllahpur, S. Suntsv, D. G. Papaglu, and S. Trtakis, Phys. Rev. Lett. 105, (010). 6. D. G. Papaglu, S. Suntsv, D. Abdllahpur, and S. Trtakis, Phys. Rev. A 81, (010). 7. N. K. Efremidis and D. N. Christdulides, Opt. Lett. 35, 4045 (010). 8. D. G. Papaglu, N. K. Efremidis, D. N. Christdulides, and S. Trtakis, Opt. Lett. 36, 184 (011). 9. P. Panagitpuls, D. G. Papaglu, A. Cuairn, and S. Trtakis, Nat. Cmmun. 4, 6 (013). 10. I. Chremms, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christdulides, and Z. G. Chen, Opt. Lett. 36, 3675 (011). 11. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christdulides, and Z. Chen, Opt. Lett. 36, 883 (011). 1. I. Kaminer, R. Bekenstein, J. Nemirvsky, and M. Segev, Phys. Rev. Lett. 108, (01). 13. E. Greenfield, M. Segev, W. Walasik, and O. Ra, Phys. Rev. Lett. 106, 1390 (011). 14. P. Zhang, Y. Hu, D. Cannan, A. Salandrin, T. Li, R. Mrandtti, X. Zhang, and Z. Chen, Opt. Lett. 37, 80 (01). 15. Y. Hu, D. Bngivanni, Z. Chen, and R. Mrandtti, Phys. Rev. A 88, (013). 16. I. D. Chremms, Z. Chen, D. N. Christdulides, and N. K. Efremidis, Phys. Rev. A 85, 0388 (01). 17. G. A. Sivilglu and D. N. Christdulides, Opt. Lett. 3, 979 (007). 18. J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987). 19. M. W. Beijersbergen, R. P. C. Cerwinkel, M. Kristensen, and J. P. Werdman, Opt. Cmmun. 11, 31 (1994). 0. M. A. Bandres and M. Guiar-Sicairs, Opt. Lett. 34, 13 (009). 1. Y. Zhang, M. R. Belić, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, Opt. Express, 7160 (014).. H. Zhng, Y. Zhang, M. R. Belić, C. Li, F. Wen, Z. Zhang, and Y. Zhang, Opt. Express 4, 7495 (016).