Programming of inhomogeneous resonant guided wave networks

Transcription

1 Programmng of nhomogeneous resonant guded wave networks Eyal Fegenbaum, 1,* Stanley P. Burgos, 1 and Harry A. Atwater 1 1 Thomas J. Watson Laboratory of Appled Physcs, Calforna Insttute of Technology, Pasadena, CA *eyalf@caltech.edu Abstract: Photonc functons are programmed by desgnng the nterference of local waves n nhomogeneous resonant guded wave networks composed of power-splttng elements arranged at the nodes of a nonunform wavegude network. Usng a compact, yet comprehensve, scatterng matrx representaton of the network, the desred photonc functon s desgned by fttng structural parameters accordng to an optmzaton procedure. Ths desgn scheme s demonstrated for plasmonc dchroc and trchroc routers n the nfrared frequency range Optcal Socety of Amerca OCIS codes: ( ) Optcal materals; ( ) Metamaterals; ( ) Plasmoncs. References and lnks 1. E. Yablonovtch, Photonc crystals: semconductors of lght, Sc. Am. 285(6), 46 51, (2001). 2. J. D. Joannopoulos, S. G. Johnson, J. N. Wnn, and R. D. Meade, Photonc Crystals: Moldng the Flow of lght, 2 nd Ed. (New Jersey, Prnceton, 2008). 3. M. Fujta, S. Takahash, Y. Tanaka, T. Asano, and S. Noda, Smultaneous nhbton and redstrbuton of spontaneous lght emsson n photonc crystals, Scence 308(5726), (2005). 4. J. Scheuer, W. M. Green, G. A. DeRose, and A. Yarv, Lasng from a crcular Bragg nanocavty wth an ultra small modal volume, Appl. Phys. Lett. 86(25), (2005). 5. D. R. Smth, J. B. Pendry, and M. C. K. Wltshre, Metamaterals and negatve refractve ndex, Scence 305(5685), (2004). 6. V. M. Shalaev, Optcal negatve-ndex metamaterals, Nat. Photoncs 1(1), (2007). 7. J. B. Pendry, D. Schurg, and D. R. Smth, Controllng electromagnetc felds, Scence 312(5781), (2006). 8. E. Fegenbaum, and H. A. Atwater, Resonant guded wave networks, Phys. Rev. Lett. 104(14), (2010). 9. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Surface plasmon subwavelength optcs, Nature 424(6950), (2003). 10. E. N. Economou, Surface Plasmons n Thn Flms, Phys. Rev. 182(2), (1969). 11. B. Prade, J. Y. Vnet, and A. Mysyrowcz, Guded optcal waves n planar heterostructures wth negatve delectrc constant, Phys. Rev. B Condens. Matter 44(24), (1991). 12. R. Za, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, Geometres and materals for subwavelength surface plasmon modes, J. Opt. Soc. Am. A 21(12), (2004). 13. J. A. Donne, H. J. Lezec, and H. A. Atwater, Hghly confned photon transport n subwavelength metallc slot wavegudes, Nano Lett. 6(9), (2006). 14. E. Fegenbaum, and M. Orensten, Modelng of Complementary (Vod) Plasmon Wavegudng, J. Lghtwave Technol. 25(9), (2007). 15. G. Verons, and S. Fan, Bends and spltters n metal-delectrc-metal subwavelength plasmonc wavegudes, Appl. Phys. Lett. 87(13), (2005). 16. E. Fegenbaum, and M. Orensten, Perfect 4-way splttng n nano plasmonc X-junctons, Opt. Express 15(26), (2007). 17. B. E. Lttle, S. T. Chu, H. A. Haus, J. Fores, and J. P. Lane, Mcrorng resonator channel droppng flters, J. Lghtwave Technol. 15(6), (1997). 18. S. Fan, P. Vlleneuve, J. Joannopoulos, and H. Haus, Channel drop flters n photonc crystals, Opt. Express 3(1), 4 11 (1998), S. E. Kocabas, G. Verons, D. Mller, and S. Fan, Transmsson Lne and Equvalent Crcut Models for Plasmonc Wavegude Components, IEEE J. Sel. Top. Quantum Electron. 14(6), (2008). 20. E. D. Palk, Handbook of optcal constants of solds, 2'nd Ed. (San-Dego, Academc, 1998). 1. Introducton The desgn and development of new photonc components s at the heart of the electrooptcs feld. Artfcal photonc materals allow for the possblty of devce desgns based on new (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25584

2 underlyng physcs. One way of classfyng artfcal photonc desgns s by ther basc resonatng elements wth whch lght nteracts to gve the desred artfcal dsperson. Under ths classfcaton scheme, we can thnk of materals that operate based on the nonlocal nterference of perodc waves (.e. photonc crystals) and of materals n whch the waves nteract locally wth subwavelength resonatng elements (.e. metamaterals). The dscovery of photonc crystals [1,2] led to extremely hgh qualty factor cavtes usng all delectrc materals based on the dea of Bragg wave physcs [3,4]. The feld of metamaterals [5,6] enabled the exctng research area of transformaton optcs [7] based on the ablty to locally tune the materal electromagnetc propertes. Dfferent n prncple from these exstng concepts, resonant guded wave networks (RGWNs) were recently reported as a new class of artfcal materal [8] where localzed waves resonate n closed paths throughout a network of solated wavegudes connected by wave splttng elements. The resultng multple resonances wthn the network gve rse to wave dsperson that s senstve, and thus tunable, accordng to the network layout. Here we utlze these dstnctve RGWNs propertes to formulate a new method for desgnng photonc components. Fg. 1. RGWNs bascs: schematc llustraton of (a) 4-termnals equal power splttng element and (b) local resonance formng n a 2x2 RGWN constructed from 4 splttng elements and 4 solated wavegudes. (c) Plasmoncs mplementaton of RGWN as ntersectng ar gaps n Au matrx: (I) X-juncton as power splttng element (II) llustraton of two wave splts acqurng dfferent phase/ampltude changes as they follow dfferent trajectores n the network and then nterfere wth each other. (d) Illustraton of possble nhomogeneous RGWN layout. A RGWN s comprsed of power splttng elements connected by solated wavegudes. The functon of the splttng element s to dstrbute a wave enterng any of ts termnals between all of ts termnals, as llustrated n Fg. 1a. The waves are then propagated n solated wavegudes between the splttng elements, where the local waves from dfferent wavegudes are coupled. Four splttng elements arranged n a rectangular network layout form a 2x2 RGWN (see Fg. 1b). When one of the termnals s excted, the mult splttng occurrences of the ncdent wave wthn the network form closed path resonances that reshape the dsperson of the emergng waves accordng to the network layout and s dfferent from the dsperson of the ndvdual wavegudes. Furthermore, the nterference of the local waves, for a properly desgned network layout, consderably enhances the resonance qualty factor due to the contrbuton of coherence effects [8]. The 2x2 RGWN supports one closed loop resonance, however larger two- or three-dmensonal networks can support multple resonances that enrch the network dynamcs. (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25585

3 One possble mplementaton of a RGWN s by usng plasmoncs va a mesh of ntersectng sub-wavelength ar gaps n a metal matrx as llustrated n Fg. 1c. Surface plasmon polartons (referred here as plasmons for brevty) are slow surface waves propagatng on metal-delectrc nterfaces [9,10]. Metal-nsulator-metal (MIM) wavegudes support a hghly confned plasmon wave (TM 0 mode) that does not get structural cutoff as the delectrc gap between the metal layers becomes vanshngly small [11 14], allowng for such plasmonc components as hgh transmsson sharp wavegude bends and power spltters [15]. In ths mplementaton, the ntersecton of two sub-wavelength MIM wavegudes forms an X- juncton that functons as the power splttng elements n the network [8,16] (see llustraton I n Fg. 1c), and the MIM segments between the ntersectons serve as the solated wavegudes connectng the X-junctons. As the ar gap thckness s vared the power splttng between the termnals can be tuned both n terms of ampltude and phase. Ths, n addton to determnng the phase accumulaton n the wavegude segments, set ndependent controls over the desgn of the nterference pattern n the RGWN. The utlty of these degrees of freedom was demonstrated n the desgn of artfcal dsperson n homogeneous RGWNs desgned to have a frequency band gap at nfrared frequences, where the metal dsperson s neglgble [8]. Because the underlyng physcs of RGWNs s based on the nterference of local waves, t allows for layouts that are nhomogeneous and non-perodc across the network. Unlke photonc crystals, whch are restrcted to Bragg wave effects n perodc structures, the flexblty of RGWNs open up possbltes for desgns where the wave propertes are vared across the structure. Wth respect to metamaterals, whch could nherently be nonhomogeneous due to the local nature of the nteracton between lght and the meta-atoms, RGWNs have the advantage of havng nterference effects wthn the network, whch allows for frequency spectrum reshapng desgns through nterference effects. An addtonal unque feature of RGWNs relates to the constrants on wave propagaton wthn the structure. Unlke other photonc desgns, RGWNs have a lmted number of modes that are allowed to propagate wthn the structure (e.g., TM 0 mode). Furthermore, the waves can propagate only nsde the wavegudes connectng the splttng elements. The dfferent wavegudes are coupled only n the X-junctons, whch each have only a lmted number of termnals. Ths level of control s benefcal for several reasons. Frst, the nterference pattern n the network can be controlled more drectly. Schematcally, there s complete control over the combnaton of the local waves that nterfere as well as the phase and the ampltude that they gan n ther dfferent trajectores, as llustrated n Fg. 1c (II). Second, t allows for a comprehensve mathematcal representaton of the RGWN by scatterng matrx (S-matrx) formalsm whch reduces the computatonal complexty of programmng the network, as demonstrated below. Thrd, snce the wavegudes are solated from each other, ther only contrbuton to the network s to serve as phase retardaton elements between the splttng elements. As a result, the wavegude length s the only effectve parameter n ts contour, as long as the bendng s not too severe, as s the case for MIM wavegudes. Ths feature of the wavegudes allows for the network to mantan ts engneered functon even when dstorted, as schematcally llustrated n Fg. 1d. The ablty to utlze curved or bent wavegudes to accommodate long contours s useful when desgnng the nterference pattern of RGWNs. These dstnctve RGWN characterstcs open up new opportuntes n desgnng photonc devces by programmng the entre network rather than by assemblng nterconnected dscrete components wth traceable functons. The usual way of desgnng photonc devces s to target the desred subsystem functons, map them logcally nto sub-functons, and then assemblng components that carry out these sub-functons nto the desred system. For example, a wavelength router could be desgned usng add/drop ports where the nput and output wavegudes are coupled by wavelength senstve rng resonators [17] or by defects n a photonc crystal [18]. Smlarly, n free space optcs, ths functon could be acheved through the use of collnear beam spltters, each desgned to deflect a desred wavelengths band. In these schemes, the couplers and wavegudes are dscrete components that are assocated wth a specfc functon, and are combned n a logcal way to carry out the overall system functon. An alternatve approach s to use a network of components that carres out the desred (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25586

4 functon but, unlke tradtonal desgns, there s no specfc logcal sub-functon assocated wth any ndvdual component. Whle the nner connectvty of the devce wll be less ntutve, t has the potental to result n effcent desgns of complex and compact devces. In ths Letter we exemplfy ths desgn paradgm by programmng a RGWN to functon as a color router. 2. Programmng RGWN usng S-matrx formalsm One possble way of representng system functon n a RGWN s through the use of a scatterng operator that maps the set of local waves enterng the devce termnals to the set of the waves extng from the same termnals [19]. Snce a RGWN s composed of a dscrete set of components (wavegudes and X-junctons) and termnals, the system functon s represented by a scatterng matrx (S-matrx) connectng the vectors of the waves enterng and emergng from the RGWN va the external ports (see Fg. 2a). Desgnng the system functon of the RGWN s then mathematcally equvalent to desgnng the S-matrx to yeld a desred output, gven a set of nputs. To elucdate the desgn prncple, programmng wll frst be demonstrated for a 2x2 RGWN. Ths network contans four wavegudes ntersectng each other n four X-junctons arranged n a 2x2 array. The wavegudes are mplemented here as ar vods n a Au matrx and could dffer n wdth, length, and contour. The devce has eght termnals, numbered from 1 to 8, as llustrated n Fg. 2a. The nput vector lsts the complex ampltudes of the magnetc felds (H-felds) enterng the network n the eght termnals, and smlarly the output vector descrbes the complex H-feld ampltudes of the waves extng the network through these same termnals. Fg. 2. Mathematcal representaton reference of (a) 2x2 RGWN system and ts components, (b) wavegude component (c) X-juncton component The network S-matrx s assembled from the mathematcal representaton of ts components accordng to the network layout. As a frst step, a functon lbrary of mathematcal representatons s generated for all the possble network components (.e., wavegudes and X-junctons) usng fnte dfference tme doman (FDTD) full wave electromagnetc smulatons. Once ths lbrary s establshed, the RGWN S-matrx can be assembled accordng to ts layout. Once the component lbrary s establshed, the S-matrx calculaton scheme s almost always much faster than resolvng the RGWN behavor from full wave electromagnetc smulatons, yet reproduces the same nformaton about the network. Ths becomes sgnfcant for optmzaton tasks and especally as the network sze ncreases. (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25587

5 Fg. 3. Tunabllty of the wave propertes n RGWN components. TM 0 modal effectve ndex n an MIM wavegude plotted versus ts wdth and exctaton frequency - (a) Real and (b) magnary parts. Transmsson coeffcents to the dfferent termnals when wave power splttng occurs at an X-juncton (c) ampltude and (d) phase of the transmsson coeffcent. The varables wdth1 and wdth2 correspond to the ar gap wdths of two normally ntersectng MIM wavegudes at an X-juncton. Two-dmensonal contours of these plots can be found n appendx A. The RGWN contans two types of components, wavegudes and X-junctons, whch need to be represented mathematcally. The wavegudes are mathematcally represented by ther complex phase retardaton, determned by the complex propagaton constant of the wave and the length of the wavegude. The propagaton constants are extracted from FDTD smulatons of wavegudes wth varous wdths at dfferent frequences, excted by a contnuous wave (CW) modal source. Fgures 3a and 3b depct the real and magnary parts of the modal effectve ndex, respectvely, where n eff = β/k 0 (the rato of the wave propagaton coeffcent to the free space wavenumber). Fgure 3a llustrates that as the wavegude sze s reduced, the effectve ndex ncreases, wth no sgnfcant dfference across the spectral range of nterest. Fgure 3b shows that Ohmc loss n the wavegudes wll cause modal attenuaton wth decay length of tens of mcrons n the near nfrared. However, as long as the attenuaton length s substantally larger than the closed path of the domnant resonances, the RGWN functon wll not be sgnfcantly altered as demonstrated here and shown n [8]. Throughout ths Letter, the Au s materal modeled accordng to tabulated data of ts optcal coeffcents [20]. The other component of the RGWN s the X-juncton, comprsed of two ntersectng wavegudes wth four termnals, whch s mathematcally represented by a (4x4) S-matrx. For a gven frequency, snce the ntersecton of two wavegudes s defned by a small number of structural parameters, namely the wavegude wdths and ntersectng angle, the parameter space for defnng the complete lbrary of X-juncton components s greatly reduced. To further smplfy our analyss, we set here the two wavegudes to be normally ntersectng (.e. settng the ntersecton angle to 90 degrees). For a gven set of wavegude wdths, the complex transmsson coeffcents of the X-juncton ports are extracted from FDTD smulatons by measurng the ampltude and phase of the wave transmtted to the dfferent ports when excted (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25588

6 from one of the termnals at a gven wavelength. The complex transmsson coeffcents nto the forward, sdeways, and backwards arms are denoted by t F, t S, t B, respectvely. Because of the symmetry of the two normally ntersectng wavegudes, the transmsson coeffcents of the two sde ports are the same. Fgures 3c and 3d plot the ampltude and the phase of the transmsson coeffcents as a functon of the two wavegudes wdths at wavelength of 1.5 μm. When the wavegudes wdths both equal 200 nm, the ncdent wave power s splt equally (relatve feld ampltude equals 0.5) and the reflected wave s π phase-shfted from the forward and sdeways waves. Increasng the two wavegudes wdths together (.e., d 1 =d 2 ) devates from the equal splt as more power s transmtted forward; however, havng dfferent wdths for the two wavegudes expands the scope of possble mathematcal representaton of components as llustrated n Fg. 3. The S-matrx of the 2x2 RGWN s assembled from the mathematcal representaton of components accordng to the network layout. The explct S-matrx representaton s gven n appendx B. We valdated the complex feld predctons of the S-matrx representaton wth FDTD smulatons, fndng less than 5% dfference for varous test cases. The two major contrbutons to the small devaton result from the nterpolaton between the parameter space ponts, where the lbrary components were calculated, and from the error added when the wavegudes are bent. For cases where no nterpolaton or wavegude bendng occurs, the FDTD results dffer only by 1% from the S-matrx predctons. The ablty to accurately predct the RGWN nterference usng S-matrx representaton allows desgnng t by an effcent optmzaton scheme. Programmng the RGWN s then performed by mnmzng the dfference between the network output and the desred one (for a gven nput) by sweepng the network parameter space, whch ncludes the wavegude wdths and lengths. The optmzaton process wll result wth a set of network parameters that can be translated to a network layout and then valdated wth FDTD smulatons. 3. Desgnng a color router We exemplfy the S-matrx programmng method by desgnng a 2x2 RGWN to functon as a dchroc router (Fg. 4a). Although smple n concept, settng a passve devce to have dfferent functons at dfferent wavelengths s qute nstructve. Explctly, the requred functon s to route two dfferent wavelengths (λ 1 and λ 2 ) to a dfferent set of ports ( 1 and 6 for λ 1 and 2 and 5 for λ 2 ) when the two bottom ports ( 7 and 8 ) are smultaneously excted wth equal power. Mathematcally, we can represent the devce as an 8x8 S-matrx S(λ 1, λ 2 ) connectng the nput and the output vectors. For both wavelengths, the nput vector s nonzero for the bottom ports (.e. In = (0,0,0,0,0,0,1,1)) and the desred output vectors would be Out(λ 1 ) = (1,0,0,0,0,1,0,0) for λ 1 and Out(λ 2 ) = (0,1,0,0,1,0,0,0) for λ 2. Because we do not have enough degrees of freedom n ths small 2x2 network to exactly attan the desred outputs, we optmze the rato of power gong to the two sets of ports at the dfferent wavelengths. Full detals of the optmzaton procedure can be found n Appendx C. After defnng the optmzaton functon we constran the parameter space based on practcal consderatons. The parameter space ncludes the wdth and length of the upper, lower, and sde wavegudes as well as the two wavelengths of operaton (λ 1 and λ 2 ). We decrease the number of parameters to optmze by restrctng the devce to have left-rght symmetry based on the desred operaton. We restran the desgn to operate n the nfrared frequency range (λ 0 = 1.2 μm 2 μm) where the materal dsperson and loss are less pronounced than n the vsble. Furthermore, the wavegude thckness s constraned to be small enough to only support the lowest order plasmonc mode (d 1,2 = 100 nm nm). The optmzaton procedure yelds the network parameters, gven n Table 1, whch reveal that the requred RGWN for color routng s dstrbuted nhomogeneously. When translatng the optmzaton parameters nto layout, we learn that the upper wavegude s longer than the lower one, and therefore needs to be bent. Importng the resultng layout nto FDTD, we obtan the steady state H-feld dstrbuton shown n Fg. 4b and 4c whch show tme snap shots for the two operaton wavelengths. A buldup of local resonance, whch results n the (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25589

7 flterng out of the desred outputs, s evdent from the temporal dynamcs of the felds, as s llustrated n Meda 1 and Meda 2. We note that the transmsson ( 3 and 4 ) and reflecton ( 7 and 8 ) ports from the devce are not dentcally zero snce the devce does not have enough degrees of freedom and therefore are not ncluded n the optmzaton functon. The FDTD smulaton results valdate the S-matrx desgn, wth λ 1 and λ 2 clearly routed to a dfferent set of sdeways ports as llustrated n Fg. 4b and 4c, respectvely. Fg. 4. 2x2 RGWN programmed to functon as a dchroc router: (a) schematc drawng, and snapshots of the H-feld at the two operaton frequences: (b) λ 1 (Meda 1) and (c) λ 2 (Meda 2). The moves show the rght half of the RGWN. Table 1. Dchroc router set of parameters for 2x2 RGWN Wavegudes Wdth Length Lower Sde Upper λ 1 = 2μm, λ 2 = 1.26μm All the dmensons n the table are gven n mcrons The matrx representaton can also be used to understand the condtons for whch the RGWN fulflls ts desred functon. From the known nput vector and the network S-matrx, the wave complex ampltudes can be dentfed at any pont n the network. For each wavelength, we resolve the exctaton condtons of the X-junctons that have the ports that are to be fltered out. For example, for λ 1 to be fltered out from termnals 2 and 5, we examne the exctaton condtons n X-juncton 3, whch has four termnals. Two of the termnals are external devce ports ( 4 and 5) and the other two are nternal network termnals. There s no nput sgnal ncdent on the two external ports, so t s the exctaton condtons of the remanng two juncton termnals that null the output n termnal 5. Indeed, the exctaton ampltudes of juncton 3 obtaned from the S-matrx representaton are 0.23exp{-j0.21π} and 0.34exp{j0.64π}, whch are close n ampltude and ~π phase-shfted. Ths s consstent wth the results from [8], whch show that when an X-juncton s smultaneously excted π phaseshfted from two adjacent termnals, the two other termnals wll be fltered out. The exctatons are not smply the same n ampltude and π phase-shfted, whch s attrbuted to the addtonal constrants the desgn has on the other wavelength as well as the lmtatons mposed on the parameter space. Smlarly, the exctaton condtons necessary for flterng out termnals 1 and 6 at λ 2 (Fg. 4b) are examned by focusng on the S-matrx ampltudes of X-juncton 4. In ths case there are three termnals beng excted: the lower termnal of the X-juncton (port 7 ) s gven by the network exctaton, so the exctaton of the other two nternal ports wll determne the flterng out of port 6. Intutvely, the condton to flter out termnal 6 wll be smply a π phase-shfted exctaton of the upper and lower termnals of juncton 4, wth zero exctaton from the sde port. From the case of λ 1 we also know that addtonal constrants mght cause a resdual wave emergng from termnal 6, whch could be compensated by a small ampltude exctaton at the other sde termnal of the juncton 4. Indeed, the exctaton ampltudes of (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25590

8 juncton 4 n the S-matrx representaton are 1 n lower termnal, 0.9exp{j0.82π} n upper termnal, and 0.3exp{-j0.32π} n the sde termnal, whch s consstent wth the ntutve pcture and s evdent from Meda 1 and Meda 2. Fg. 5. 3x3 RGWN programmed to functon as a trchroc router. Snapshots of the H-feld at the three operaton frequences: (a) λ 1 (Meda 3), (b) λ 2 (Meda 4) and (c) λ 2 (Meda 5). The moves show the rght half of the RGWN. Table 2. Trchroc router set of parameters for 3x3 RGWN Wavegudes Wdth Length Vertcal center-bottom (C-B) sde-bottom (S-B) center-top (C-T) sde-top (S-T) Horzontal Top Mddle Down λ 1 = 1.59μm, λ 2 = 1.97μm, λ 3 = 1.23μm All the dmensons n the table are gven n mcrons To further exemplfy the programmablty of RGWNs va S-matrx formalsm, we consder a 3x3 RGWN programmed to functon as a trchroc router. In order to mplement the more complex task of routng three wavelengths we allow for more degrees of freedom n the network by ncreasng the number of components, effectvely ncreasng the amount of data contaned. The functon s defned as an extenson of the dchroc router, but here when the three bottom termnals are smultaneously excted at three dfferent frequences, the frequences are fltered out to three dfferent sets of sde termnals as llustrated n Fg. 5. The analyss results n the optmal RGWN parameters shown n Table 2. FDTD smulatons valdate the resultng layout desgn, as shown n Fg Concluson In ths Letter we have mplemented a scheme for programmng resonant guded wave networks by explotng ther unque nterference propertes. The constrants on the propagaton and couplng of the local waves n RGWNs allow for the devce operaton to reduce to a smple mathematcal representaton usng S-matrx formalsm. Ths allows for the network programmng to take the form of an optmzaton procedure over a relatvely small parameter space. The RGWN S-matrx representaton was demonstrated here where the nputs are gven and the S-matrx of the devce s desgned to gve the desred outputs (e.g., routng, modal converters). However, ths formalsm could be extended to dfferent type of functons, such as sensng, n whch the nputs are gven and the output changes are montored. The specfc RGWN functon examned here s color routng, whch s challengng n terms of havng to program a dfferent functon for dfferent wavelengths all wthn the same (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25591

9 devce. Even though t was meant as a concept demonstraton, ncorporatng more components nto the RGWN and therefore ncreasng the possble degrees of freedom, may evolve to a more complex devce or alternatvely a devce wth enhanced performance. Appendx A. Tunabllty of the wave propertes n RGWN components. The three-dmensonal data of Fg. 3 s plotted n Fg. A1 n two-dmensonal contours to allow for easer read n a quanttatve manner. Fg. A1. Tunablty of the wave propertes n RGWN components. (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25592

10 Appendx B. S-matrx representaton of 2x2 RGWN Ths appendx descrbes the assembly of the 2x2 RGWN S-matrx from the propertes of ts components accordng to the network layout. The phasor representaton of the local wave H-felds n the network s represented by three column vectors (transposed for brevty): tr o o o o o o o o out 1(1) 1(2) 3(3) 3(2) 2(1) 2(4) 4(3) 4(4) tr n 1(1) 1(2) 3(3) 3(2) 2(1) 2(4) 4(3) 4(4) tr net 1(3) 1(4) 3(1) 3(4) 3(3) 3(2) 4(1) 4(2) A a, a, a, a, a, a, a, a ; A a, a, a, a, a, a, a, a ; A a, a, a, a, a, a, a, a ; (B-1) A out and A n hold the values of the local nput and output waves of the RGWN at ts ports, and A net represents the nput wave on the X-junctons from the nternal termnals of the RGWN. The /o denotes nput/output waves n respect to the X-juncton, the junctons are numbered as n Fg. 2a, and the juncton ports are numbered n brackets, n accordance to Fg. 2c. The couplng of the H-feld vectors by the network connectvty s represented wth the system: Aout M A FS net M A RS n 0 M K RS Anet M A FS n (B-2) where M FS and M RS are dagonal 8-by-8 matrces that orgnate from the splttng relatons n the X-junctons and K s a sparse 8-by-8 matrx that stands for the wave propagaton n the wavegudes. These matrces are defned as: FS FS ,,,,,,, FS FS FS FS RS RS RS RS RS V H V H tf ts tr t S S V H RS V H ts tf ts tr M Dag S S S S M Dag S S S S S 3 K(1,3) K(3,1) 1 2 K(2,6) K(6,2) 1 4 K K(4,8) K(8,4) 3 4 K(5,7) K(7,5) 2 other matrx elements 0 m, exp j L m (B-3) where the V/H ndex denotes f the transmsson coeffcent s for exctaton from the vertcal or the horzontal wavegude of that X-juncton. Algebra of equaton set B-2 gves the matrx representaton of the 2x2 RGWN S-matrx: 2x2RGWN Appendx C. Optmzaton procedure 1 S M M M K M (B-4) RS FS RS FS The optmzaton procedure s mplemented n Matlab usng the pre-calculated mathematcal representaton data set of the RGWN components obtaned from full-feld electromagnetc FDTD smulatons excted wth contnuous wave sources. The procedure s llustrated wth the followng flow chart n Fg. C1. (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25593

11 FDTD Pre-calculaton of component lbrary: 1. MIM effectve ndex 2. X-juncton transmsson coeffcent Network parameters Matlab 2x2 RGWN S-matrx Optmzaton functon Outputs FDTD Reconstructon of the 2x2 RGWN wth the output parameters. Fg. C1. Flow chart of the optmzaton procedure FDTD s used to calculate the complex effectve ndex of Au/Ar/Au MIM wavegudes wth ar gap thckness rangng from 100 nm to 520 nm n steps of 10 nm. The effectve ndex s calculated from the mode feld attenuaton and ts phase retardaton relatve to the speed of lght. The S-matrx data s calculated for X-junctons of two MIM ntersectng wavegudes wth ar gap thckness rangng from 100 nm to 520 nm n steps of 10 nm (wth the two wavegudes not necessarly havng the same wdth). The transmsson coeffcents are calculated by montorng the relatve feld transmtted to each of the ports relatve to the ncdent feld through any one of ts ports. The dchroc router network s defned by eght parameters: the length and wdth of the upper, lower and sde wavegudes and the two wavelengths. The wavegude wdths determne the effectve ndex n the wavegudes as well as the transmsson coeffcents of the X- junctons. The optmzaton procedure s conducted usng Matlab fmcon functon usng nteror pont settngs and a random set of 200 ntal ponts over the parameter space to reduce local mnmum pont effects. The man scrpt starts at an ntal pont wthn the parameter space and use the fmcon functon to move n the parameter space towards mnmzaton of the optmzaton functon. The optmzaton functon for the dchroc router s defned as follows: O Out ( ) / Out ( ) O Out ( ) / Out ( ) O1 O 2 Out6( 1 ) Out5( 2) O1 O2 1 f O1 O2 (C-1) where O 1 and O 2 represent the two termnals output ratos that need to be maxmzed at the two dfferent wavelengths, λ 1 and λ 2. The f functon s used to merge the two ratos together nto one weghted optmzaton parameter, where the frst term n squared brackets maxmzes the total power routed nto the selected termnals, the second term maxmzes the two ratos, and the thrd term s a weghtng factor that ensures that the two ratos are maxmzed equally. The target functon s defned as the nverse of these three terms n multplcaton snce the fmcon functon s a mnmzaton functon. At each pont n the parameter space, the network output vector s calculated as the multplcaton of the 2x2 RGWN S-matrx evaluated at the parameter values tmes the nput vector representng exctaton only from termnal 8. The optmzed network parameters are gven n Table 1 and the correspondng output feld ampltude values are gven n Table C1. (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25594

12 Table C1. Ampltudes of output termnals of the Dchroc router Termnal number λ1 λ O 1 = 4.3 O 2 = 24.5 For the Trchroc functon the optmzaton functon s: O Out ( ) / Out ( ) Out ( ) O Out ( ) / Out ( ) Out ( ) O Out ( ) / Out ( ) Out ( ) Out9( 1 ) Out8( 2) Out7( 3) O1 O2 O3 f O1 O 2 O1 O 3 O2 O O1 O 2 O1 O 3 O2 O 3 (C-2) The logc of constructng the functon s smlar to Table C1 but for weghtng the three ratos to be maxmzed. The target functon n (C-2) s optmzed yeldng the parameter values gven n Table 2. The output feld ampltude values for the trchroc router are gven n Table C2. Table C2. Ampltudes of output termnals of the Trchroc router Termnal number λ O 1 = 2.1 λ O 2 = 4.7 λ O 3 = 10.5 Acknowledgments We wsh to acknowledge the support from the Offce of Basc Energy Scences under Contract No. DE-FG02-07ER (C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 25595