Rabi oscillations and stimulated mode conversion on the subwavelength scale

Transcription

1 Rabi oscillatios ad stiulated ode coversio o the subwavelegth scale Xiao Zhag, 1,2 Fagwei Ye, 1,2,* Yaroslav V. Kartashov, 3 ad Xiafeg Che 1,2 1 Departet of Physics ad Astrooy, Shaghai Jiao Tog Uiversity, Shaghai , Chia 2 Key Laboratory for Laser Plasa (Miistry of Educatio), IFSA Collaborative Iovatio Ceter, Shaghai Jiao Tog Uiversity, Shaghai , Chia 3 Istitute of Spectroscopy, Russia Acadey of Scieces, Troitsk, Moscow Regio , Russia * fagweiye@sjtu.edu.c Abstract: We study stiulated ode coversio ad dyaics of Rabi-like oscillatios of weights of guided odes i deeply subwavelegth guidig structures, whose dielectric perittivity chages periodically i the directio of light propagatio. We show that despite strog localizatio of the fields of eigeodes o the scales below the wavelegth of light, eve weak logitudial odulatio couples odes of selected parity ad causes periodic eergy exchage betwee the, thereby opeig the way for cotrollable trasforatio of the iteral structure of subwavelegth beas. The effect is reiiscet of Rabi oscillatios i ultilevel quatu systes subjected to the actio of periodic exteral fields. By usig rigorous uerical solutio of the full set of the Maxwell's equatios, we show that the effect takes place ot oly i purely dielectric, but also i etallic-dielectric structures, despite the eergy dissipatio iheret to the plasoic waveguides. The stiulated coversio of subwavelegth light odes is possible i both liear ad oliear regies Optical Society of Aerica OCIS codes: ( ) Subwavelegth structures; ( ) Ihoogeeous optical edia Refereces ad liks 1. I. I. Rabi, O the process of space quatizatio, Phys. Rev. 49(4), (1936). 2. X. G. Zhao, G. A. Georgakis, ad Q. Niu, Rabi oscillatios betwee Bloch bads, Phys. Rev. B Codes. Matter 54(8), R5235 R5238 (1996). 3. M. C. Fischer, K. W. Madiso, Q. Niu, ad M. G. Raize, Observatio of Rabi oscillatios betwee Bloch bads i a optical potetial, Phys. Rev. A 58(4), R2648 R2651 (1998). 4. J. N. Wi, S. Fa, J. D. Joaopoulos, ad E. P. Ippe, Iterbad trasitios i photoic crystals, Phys. Rev. B 59(3), (1999). 5. R. W. Robiett, Quatu wave packet revivals, Phys. Rep. 392(1-2), (2004). 6. D. Marcuse, Theory of Dielectric Optical Waveguides (Acadeic, 1991). 7. K. O. Hill, B. Malo, K. A. Vieberg, F. Bilodeau, D. C. Johso, ad I. Skier, Efficiet ode coversio i telecouicatio fiber usig exterally writte gratigs, Electro. Lett. 26(16), (1990). 8. K. S. Lee ad T. Erdoga, Fiber ode couplig i trasissive ad reflective tilted fiber gratigs, Appl. Opt. 39(9), (2000). 9. Y. V. Kartashov, V. A. Vysloukh, ad L. Torer, Resoat ode oscillatios i odulated waveguidig structures, Phys. Rev. Lett. 99(23), (2007). 10. K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, ad D. Kip, Optical trasitios ad Rabi oscillatios i waveguide arrays, Opt. Express 16(14), (2008). 11. K. Shadarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, ad M. Segev, Experietal observatio of Rabi oscillatios i photoic lattices, Phys. Rev. Lett. 102(12), (2009). 12. C. D. Poole, C. D. Towsed, ad K. T. Nelso, Helical-gratig two-ode fiber spatial-ode coupler, J. Lightwave Techol. 9(5), (1991). 13. D. McGloi, N. B. Sipso, ad M. J. Padgett, Trasfer of orbital agular oetu fro a stressed fiberoptic waveguide to a light bea, Appl. Opt. 37(3), (1998). 14. C. N. Alexeyev ad M. A. Yavorsky, Geeratio ad coversio of optical vortices i log-period helical core optical fibers, Phys. Rev. A 78(4), (2008). 15. C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapi, ad M. A. Yavorsky, Geeratio of optical vortices i layered helical waveguides, Phys. Rev. A 83(6), (2011). 16. Y. V. Kartashov, V. A. Vysloukh, ad L. Torer, Dyaics of topological light states i spiralig structures, Opt. Lett. 38(17), (2013) OSA 9 Mar 2015 Vol. 23, No. 5 DOI: /OE OPTICS EXPRESS 6731

2 17. G. K. L. Wog, M. S. Kag, H. W. Lee, F. Biacalaa, C. Coti, T. Weiss, ad P. St. J. Russell, Excitatio of orbital agular oetu resoaces i helically twisted photoic crystal fiber, Sciece 337(6093), (2012). 18. V. S. Shchesovich ad S. Chávez-Cerda, Bragg-resoace-iduced Rabi oscillatios i photoic lattices, Opt. Lett. 32(13), (2007). 19. B. Terhalle, A. S. Desyatikov, D. N. Neshev, W. Krolikowski, C. Dez, ad Y. S. Kivshar, Dyaic diffractio ad iterbad trasitios i two-diesioal photoic lattices, Phys. Rev. Lett. 106(8), (2011). 20. J. Takahara, S. Yaagishi, H. Taki, A. Morioto, ad T. Kobayashi, Guidig of a oe-diesioal optical bea with aoeter diaeter, Opt. Lett. 22(7), (1997). 21. Q. Xu, V. R. Aleida, R. R. Paepucci, ad M. Lipso, Experietal deostratio of guidig ad cofiig light i aoeter-size low-refractive-idex aterial, Opt. Lett. 29(14), (2004). 22. S. Maier, Plasoics: Fudaetals ad Applicatios (Spriger, 2007). 23. F. Ye, D. Mihalache, B. Hu, ad N. C. Paoiu, Subwavelegth plasoic lattice solitos i arrays of etallic aowires, Phys. Rev. Lett. 104(10), (2010). 24. C. O. M. S. O. L. Multiphysics, 1. Itroductio Rabi oscillatios are the periodic trasitios (or revivals) betwee two statioary states of a quatu syste drive by the periodic exteral field. This is purely resoat effect, whose efficiecy strogly depeds o the frequecy detuig betwee the exteral field ad ow oscillatio frequecy of the quatu syste. Sice their predictio by I. Rabi i his seial paper [1], such revivals ad related resoat pheoea have bee studied i a variety of atoic, optical, ad codesed atter systes [2 5]. Especially fruitful was the extesio of the cocept of Rabi oscillatios to various weakly guidig paraxial optical structures where light propagatio is described by the faous Schrödiger equatio that also govers the evolutio of the wavefuctio i quatu-echaical systes [6 16]. I such paraxial structures, Rabi oscillatios aifest theselves as the periodic eergy exchage betwee liear guided odes of the ultiode structure, stiulated by weak periodic odulatio of the refractive idex i the directio of light propagatio that plays the role siilar to that of the exteral drivig field for ultilevel quatu syste. Besides purely fudaetal aspects, i optical systes the iterest to Rabi oscillatios is explaied by the potetial rich practical applicatios of this effect for the cotrollable shapig of laser radiatio. Rabi oscillatios were observed i dyaical log-periodic gratigs created i optical fibers [7,8], they were studied i various shallow ultiode waveguides [9] ad odulated periodic lattices [10,11]. Especially iterestig are realizatios of Rabi oscillatios i the two-diesioal systes, where logitudial refractive idex odulatios ay couple light odes with differet topological charges [12 17]. Notice, that soeties equatios, siilar to those describig dyaics of Rabi oscillatios, ay be ecoutered i periodic guidig structures without ay logitudial variatios [18,19]. So far, the dyaics of optical Rabi oscillatios was aalyzed ostly i the paraxial ultiode guidig structures, where all characteristic scales (bea ad waveguide widths) substatially exceed the wavelegth. I this case the coupled-ode approach applied to the Schrödiger equatio, goverig light evolutio, yields equatios for ode aplitudes aalogous to those for populatios of levels i a drive quatu syste. O the other had, last decade has witessed rapid advaces i aotechologies that already allow fabricatio of optical waveguides with subwavelegth diesios [20,21]. Cosiderable waveguide depths ca be achieved (strog guidig regie), so that eigeodes of such structures also becoe subwavelegth. Especially fruitful is the cobiatio of dielectrics ad etals allowig to cofie light due to the excitatio of surface plaso polaritos (SPPs) [22]. I such structures oe has to resort to solutio of the full set of Maxwell's equatios, sice the paraxial approxiatio fails to describe light evolutio. A geeral questio arises are Rabi oscillatios possible i the odulated subwavelegth structures ad to which extet this effect ca be used for shapig of arrow light beas? I this paper we address Rabi oscillatios i subwavelegth dielectric ad etal-dielectric waveguides ad show that this effect persists at the subwavelegth scales ad ca be used to cotrol the odal structure of the output beas. Logitudial odulatio of the dielectric perittivity i the waveguide regio results i a highly selective excitatio of odes of proper parity, while efficiecy of this process is cotrolled by the detuig of odulatio frequecy fro the resoat value. Efficiet ode coversio ca be achieved at very short propagatio distaces OSA 9 Mar 2015 Vol. 23, No. 5 DOI: /OE OPTICS EXPRESS 6732

3 2. The odel ad coupled ode theory We cosider the propagatio of a TM-polarized light bea alog the z -axis of a subwavelegth waveguide with logitudially odulated dielectric perittivity. The evolutio of ozero copoets of the electric E = ( Ex,0, Ez) ad agetic H = (0, H y, 0) fields for selected polarizatio state is govered by the reduced syste of Maxwell's equatios: Ex 1 é 1 H y ù i =- - wh z ew 0 x êëe(, x z) x úû H y i =-ee 0 (, x z) wex, z 0 y where we have excluded the logitudial copoet Ez= [/ i ee 0 (, x z) w] Hy/ x of the electric field for coveiece; e 0 ad 0 are the vacuu perittivity ad pereability; ad w is the frequecy of light. The fuctio e (, xz) describes the relative perittivity of the guidig structure. I the case of dielectric waveguide we set e = e + + d w - (2) (, xz) 2 2 bg p[1 si( zz)]exp( x / a), where e bg is the relative backgroud perittivity. The fuctio (2) describes the profile of Gaussia waveguide with width a ad depth p, which oscillates alog the propagatio directio with the aplitude d ad frequecy w z. For coveiece we deote dyaically varyig part of dielectric perittivity as de 2 2 li( xz, ) = pd si( wzz)exp( - x / a). I the absece of logitudial odulatio ( d = 0), all eigeodes propagate idepedetly ad the power cocetrated withi each ode reais uchaged. Logitudial perittivity odulatio iside the waveguide couples differet odes ad stiulates eergy exchage betwee the. Before we go to the uerical siulatios of Eq. (1) ad (2), it is istructive to cosider the dyaics of this process usig coupled-ode approach [6,23]. I this approach we use Loretz reciprocity forula ( H E * u+ H * u E) = iw( e-eu) EE * u that liks the fields { E, H } i the waveguide with logitudially odulated perittivity e= e with those { Eu, H u} i the uodulated guide with e= eu, where e- eu= deli. The field i the odulated waveguide ca be represeted as a superpositio { E, H } ( ) i z{ E, H } =å a z e b of eigeodes with z -depedet aplitudes a() z. A siilar represetatio ca be used for { Eu, H u} but with a = cost. We substitute fields i such for ito reciprocity forula, itegrate it with respect to trasverse coordiate x, ad, takig ito accout the orthogoality coditio ez ( E H ) dx = d, arrive to the coupled-ode equatios for aplitudes a () z * : ò k k da i = å w a exp[ i( b -b ) z], (3) dz where z -depedet coefficiets () z are give by: ò 2 li( * * de Ex Ex Ez Ez ) dx/ Re( Ex Hy * = + ) dx, (4) Takig ito accout variatio of deli si( w z z) alog the propagatio distace, oe ca see that eergy exchage betwee odes ad will be ost efficiet if the resoace coditio b-b wz = 0 is satisfied, provided that the odes have the sae syetry, so that correspodig itegral i the uerator of (4) is ozero. 3. Nuerical results ad discussio Rabi oscillatios occur i ultiode guidig structures. We search uerically for eigeodes of the uodulated ( d = 0) subwavelegth Gaussia waveguide i the for { E( xz, ), H ( xz, )} = ò, (1) 2015 OSA 9 Mar 2015 Vol. 23, No. 5 DOI: /OE OPTICS EXPRESS 6733

4 { E( x), H ( x)}exp( ibz). The copoets of vectors E(), x H () x describe the trasverse field distributios, while b are the propagatio costats of the odes with idexes = 1,2,.... We cosider a illustrative situatio whe the waveguide supports three guided odes. This occurs for the followig paraeters: e bg= 2.25, p= 2, ad a = 250. The width of the waveguide a is substatially saller tha the wavelegth l = The profiles of the trasverse ( E x) ad logitudial ( E z) electric field copoets i all three guided odes are show i Figs. 1(a) ad 1(b), respectively. Modes are cofied o the scale coparable with the wavelegth l. The logitudial field copoet is coparable i aplitude with the trasverse field oe. The syetry of Maxwell's equatios dictates that E z is atisyetric for syetric E x ad that the uber of odes i E z always exceeds by oe the uber of odes i E x. I the followig, whe we etio the parity of the ode, we refer to the shape of the E x copoet. The propagatio costats of the eigeodes depicted i Figs. 1(a) ad 1(b) are give by b 1» 1.919k0, b 2» 1.689k0, b 3= 1.531k 0, where k0 = w / c is the vacuu waveuber. Fro the coupled ode equatio [Eq. (3)], due to ozero coefficiet 13, the couplig is expected to occur for the first ad third odes with equal parity at wr = b1- b3» 0.389k0 that correspods to a odulatio period 2 p/ w r» Notice, that efficiet couplig betwee first ad secod odes is ipossible for syetric perittivity perturbatios e( - xz, ) = e( xz, ), sice correspodig couplig coefficiet 12 vaishes (couplig of these odes requires perturbatios that break the syetry of e, such as trasverse oscillatios of the waveguide ceter). Fig. 1. Trasverse (a) ad logitudial (b) electric field distributios i three liear eigeodes of Gaussia waveguide with p = 2 ad a = 250. Trasverse (c) ad logitudial (b) electric field distributios i three eigeodes supported by the dielectric slab with a = 600 ad p = 2 surrouded by etal with Ree =- 20. The efficiet couplig of the first ad third odes of the subwavelegth waveguide due to resoat ( wz = wr) logitudial odulatio is illustrated i Figs. 2(a) ad 2(b). The results are obtaied by direct solutio of Maxwell's Eqs. (1) with fiite-eleet ethod [24]. I Fig. 2(a) oly the first ode was provided at the iput, while i Fig. 2(b) oly the third ode was provided. Oe ca observe gradual trasforatio of the iitial bell-shaped field distributio ito three-hup distributio, characteristic for the third guided ode. I agreeet with Eqs. (3), after accuulatio of the power i the third ode, the reverse process starts ad power flows 2015 OSA 9 Mar 2015 Vol. 23, No. 5 DOI: /OE OPTICS EXPRESS 6734

5 back ito the first ode. This results i periodic Rabi oscillatios havig for d = 0.1 the period as short as 86, irrespectively of the iput ode idex. The efficiecy of ode coversio ca be quatified usig istataeous weights of the od- 2 () z = c () z, where es Fig. 2. (a),(b) Dyaic of resoat oparaxial ode coversio at d = 0.1 i dielectric structure at wz = wr» 0.389k0, whe oly first or third ode is provided at the iput. The propagatio distace ad trasverse widow size are 130 ad 4, respectively. (c),(d) Dyaics of ode coversio at d = 0.1 ad wz = wr» 0.232k0 i etal-dielectric-etal structure. The propagatio distace ad trasverse widow size are 75 ad 2, respectively. Oly E 2 x distributio is show. x y * x y * - - c () z E (, x z) H dx / E H dx, =ò ò (5) are the coplex coefficiets deteriig odal copositio of the trasverse copoet of the electric field Ex( x, z ) at the distace z, while Ex ( x), H y( x ) characterize field distributios i eigeodes. The depedecies () z correspodig to dyaics fro Figs. 2(a) ad 2(b) are show i Figs. 3(a) ad 3(b). Oe ca see that decrease i 1 is accopaied by the growth of the weight of the third ode 3 ad vice versa, that idicates o resoat couplig of these two odes. At the sae tie, secod ode is ot excited ad its weight 2 reais early zero. While odal field structure is i fact recovered after oe period of oscillatios, radiatio ad backward reflectios, uavoidable for odulatio depths d ad periods 2 p/ w z cosidered here, slightly reduce the output quatity This circustace is ot accouted for by coupled-ode approach ad ca be captured oly by solutio of the full set of Maxwell's equatios. Radiative losses rapidly decrease with decrease of d. Stiulated ode coversio is the resoat effect. While i resoace early all power is trasferred betwee odes, the efficiecy of coversio drops dow with icrease of the detuig of odulatio frequecy fro the resoat oe. I Fig. 4(a) we show axial weight of the third ode ax 3 acquired upo propagatio as a fuctio of the odulatio frequecy i the case, whe oly first ode is lauched ito structure at z = 0. The resoace is syetric i w z its width depeds o the depth of the logitudial perittivity odulatio. The width of the 2015 OSA 9 Mar 2015 Vol. 23, No. 5 DOI: /OE OPTICS EXPRESS 6735

6 resoace dw z, defied at the level ax 3= 3 /2, icreases with d alost liearly [Fig. 4(b)]. For sall d values resoat excitatio of higher-order odes (i the case whe structure supports several of the) ca be ade highly selective due to sall resoace width. Aother iportat quatity that characterizes the process cosidered here is the coversio legth, or period of Rabi Fig. 3. Mode weights 2 k = ck versus propagatio distace for 1«3 ad 3«1 resoat trasitios. The depedecies () i (a),(b) correspod to evolutio dyaics show i Fig. 2. k z Fig. 4. (a) Maxial weight of the third ode versus odulatio frequecy at d = 0.1 i the liear ediu. Resoace width (b) ad effective couplig distace at wz = wr (c) versus refractive idex odulatio depth d i the liear ediu. (d) Maxial weight of the third ode versus odulatio frequecy i focusig ediu for peak oliear cotributio to the refractive idex l = 0.02 ad d = 0.1. oscillatios z c, defied here as a distace at which ost of the power returs back to the iput eigeode at wz = wr. I accordace with coupled-ode approach, where couplig distace evolves 1/ 13, with 13 d, oe observes i Fig. 4(c) rapid decrease of z c with icrease of the perittivity odulatio depth d. Oe should stress that i this structure the coplete trasforatio of the shape of subwavelegth light spots ca be achieved at propagatio distaces as short as 20, provided that the perittivity odulatio depth d is large eough OSA 9 Mar 2015 Vol. 23, No. 5 DOI: /OE OPTICS EXPRESS 6736

7 So far, Rabi oscillatios were cosidered oly i dielectric guidig structures. The questio arises whether siilar effect ca be observed i etal-dielectric waveguides, where the presece of etal allows to easily achieve light cofieet at the subwavelegth scales. We thus cosider etal-dielectric-etal waveguide, whose perittivity is give by Eq. (2) for x 300 ad by e (, xz) = i(perittivity of silver at l = ) for x > 300. The shapes of eigeodes of such a waveguide strogly differ fro those i purely dielectric structures ad the odes are ore cofied, but oe still ca see that the first ad third odes are syetric, while secod ode is atisyetric [Figs. 1(c) ad 1(d)]. We foud that despite losses i etal, the resoat logitudial perittivity odulatio i the dielectric regio still results i efficiet couplig betwee first ad third odes i such a structure, as show i Figs. 2(c) ad 2(d). Fially, Rabi oscillatios are possible i the presece of focusig oliearity i the dielectric. We accout for oliear respose of the aterial by assuig that perittivity (2) icludes 2 2 a additioal ter de l = 2( Ex + E z ), where 2 > 0 is the oliear coefficiet. Solutio of the Maxwell's Eqs. (1) reveals that weak focusig oliearity reduces the coversio legth z c [this is a direct cosequece of the icrease of the effective waveguide depth leadig to growth of the overlap itegral 13 i (4)], ad also shifts the exact resoace frequecy to a higher value [Fig. 4(d)]. The latter fact ca be explaied takig ito accout that oliearity acts differetly o propagatio costats of eigeodes, leadig to higher growth of b 1 i copariso with b 3, thereby icreasig the differece b1- b3. The oliear resoace curve becoes slightly asyetric. 4. Coclusio I this paper, we have deostrated that the Rabi oscillatios are possible i various deeply subwavelegth guidig structures (with widths dow to 100 ) icludig subwavelegth dielectric waveguides ad etal-dielectric-etal guidig structures, where the dyaics is govered by Maxwell's equatios. Rabi oscillatio for subwavelegth odes ay occur at very short propagatio distaces (about tes of icros), without cosiderable backward reflectios, i both liear ad oliear regies. This eables cotrollable shapig of deeply subwavelegth lights spots at extreely short distaces. Ackowledgets The work of X. Zhag ad F. Ye was supported by the Natioal Natural Sciece Foudatio of Chia (Grat No ad ) OSA 9 Mar 2015 Vol. 23, No. 5 DOI: /OE OPTICS EXPRESS 6737