Plasmonic Waveguide Analysis

Transcription

1 Plamoni Waveguide Analyi Sergei Yuhanov, Jerey S. Crompton *, and Kyle C. Koppenhoeer AltaSim Tehnologie, LLC *Correponding author: 3. Wilon Bridge Road, Suite 4, Columbu, O 4385, je@altaimtehnologie.om Abtrat: Thi paper ompare numerial and analyti olution or a typial plamoni waveguide oniting o a thin ilm andwihed between a ladding over and a ubtrate. Two oniguration are analyzed uing the letromagneti Wave, Frequeny Domain interae (emw) o the RF Module o COMSOL Multiphyi: dieletri-metal dieletri (DMD) and metal-dieletri-metal (MDM) layer. The analyti olution developed by Oranidi [] provide a omparion with the numerial reult. Keyword: Plamoni waveguide, eletromagneti, analyti olution.. Introdution Surae Plamon (SP) or Surae Plamon Polariton (SPP) are eletromagneti eitation that propagate at the interae between a dieletri and a ondutor, and are evaneently onined in the perpendiular diretion to the propagation. They arie via oupling o the eletromagneti ield to oillation o the ondutor eletron plama and are haraterized in term o diperion and patial proile. From an eletrodynami view, SP are a partiular ae o a urae wave: rom the opti view, SP are optial mode o an interae: rom the olid-tate phyi view, SP are olletive eitation o eletron. The behavior o Surae Plamon an be deribed by Mawell equation a long a the propertie o metal at optial requenie an be obtained rom eperiment or theoretially alulated rom the Drude model o eletron ondution. Surae plamoni waveguide have the ability to onine light at ub-wavelength ale and have a large number o appliation in the ield o nanoiruit, nanophotoni devie, biologial and hemial enor, holography, and other appliation. Ue o plamon in eletri iruit, or in an eletri iruit analog, ombine the ize eiieny o eletroni with the data apaity o photoni integrated iruit. Both urae plamon polariton propagating along the metaldieletri interae and loalized urae plamon mode upported by metal nanopartile are haraterized by large momentum value, whih enable trong reonant enhanement o the loal denity o photon tate and an be utilized to enhane weak optial eet o optoeletroni devie. Dierent plamoni waveguide truture have been propoed, uh a layered truture, metalli nanowire, metalli nanopartile array, hybrid wedge plamoni waveguide, and other oniguration. ere, a typial plamoni waveguide oniting o a thin ilm andwihed between a ladding over and a ubtrate will be onidered. Two oniguration are analyzed uing the letromagneti Wave, Frequeny Domain interae (emw) o the RF Module o COMSOL Multiphyi: dieletri-metal dieletri (DMD) and metal-dieletri-metal (MDM) layer. Reult are ompared againt an analyti olution developed by Oranidi [].. Double Interae Plamoni Waveguide Analyti Solution The waveguide geometry or the DMD and MDM oniguration i hown in Fig.. The waveguide onit o a thin ilm, andwihed between a ladding over and a ubtrate. The layer have ininite etent along the z diretion. In the DMD oniguration, the ilm layer i metal, while ubtrate and ladding are dieletri layer. In the MDM oniguration, the ilm layer i dieletri, and metal are ubtrate and ladding. Only TM plamoni mode will be onidered, although T mode are alo poible in more erpt rom the Proeeding o the 5 COMSOL Conerene in Boton

2 ompliated media, uh a metamaterial and magneti material []. Figure. Plamoni waveguide in either DMD o MDM oniguration. The analytial olution given below i taken rom tetbook by Oranidi []. In a waveguiding ytem hown in Fig., the eletri and magneti ield are propagating along the guiding diretion. The layer have ininite etent along the z diretion. Field are aumed to have the ollowing orm: jωt jβ (, y, t) = ( y) e jωt jβ (, y, t) = ( y) e () where β i the propagation wavenumber along the guide diretion. For TM mode, there i one longitudinal ield and two tranvere ield y, z. The ret omponent o the eletromagneti wave are zero. The longitudinal ield atiie the elmholtz equation () k = y The tranvere eletri ield i omputed rom Mawell equation in term o longitudinal ield: jβ (3) k y y = The tranvere magneti ield i obtained rom tranvere eletri ield and TM impedane η TM : y z β (4) z = y, ηtm = ω ηtm The uto wavenumber k appearing in the elmholtz equation depend on dieletri ontant o the propagating medium, k = k β, where k = π / λ = ω / i the vauum wavenumber and λ i the vauum wavelength, and i ree-pae peed o light. Thereore, k take dierent value in eah layer. Within the ilm with a dieletri ontant o, the tranvere uto wavenumber atiie k = k β. The ield annot penetrate into the metal and will be eentially urae wave that deay eponentially away rom the metal dieletri interae. Deining ilm attenuation oeiient by γ = jk, we obtain the ollowing relationhip within the ilm region: γ = β k, y a (5) We look or ield olution that deay eponentially away rom interae o that uto wavenumber in ubtrate and ladding region are pure imaginary. Deining tranvere attenuation oeiient by α = jk or ladding and α = jk or ubtrate, we obtain the ollowing relation: a β (6) = k, a = β k, y a y Thu, the TM mode are obtained by olving the elmholtz equation in eah layer y y y γ =, =, =, or y a or y a or y (7) erpt rom the Proeeding o the 5 COMSOL Conerene in Boton

3 quation (7) i upplemented by the ontinuity boundary ondition or the tangential ield at the layer interae: (8) =, = y= y=+ y= a y= a+ tαnh tαnh ( γα + ψ ) ( γα ψ ) α = γ α = γ () = z y= z y=+ = z y= a z y= a+ (9) Five parameter β, γ, α, α, ψ are determined rom ive tranendental algebrai equation (5), (6), (). Inide the ilm, the olution to the elmholtz equation an be ontruted uing linear ombination o hyperboli term ( γy) ( γy) inh and o - reerred to a plamoni olution. The olution that automatially atiie the ontinuity boundary ondition (8) an be epreed a ollowing: ( y) inh = inh inh ( γ y + y ), y a () ( y) ( γa + y ) e, y a a ( y+ a) ( γa y ) e, y Tranvere eletri ield i obtained rom relation (3): j y jβ = oh( γy + y ), γ γ jβ y jβ ( y) y ( y) = = inh( γa + y ) e, a jβ y jβ a ( y+ a) = inh( γa y ) e, a β () y a y a y The tranvere magneti ield i then obtained rom the relation (4), where TM impedane i deined a ηtm = β / ω or ilm layer, ηtm = β / ω or ladding layer, and = / or ubtrate. ηtm β ω The interae ontinuity ondition o the tangential -ield (9) yield the ollowing two equation: The ield amplitude an be obtained rom total power P tranmitted in plamoni waveguide. The omponent o the Poynting vetor give the power low in the diretion per unit area: P = Re [ ( y) ( y) ]= y [ η ] ( y) Re / TM y z (3) Integrating (3) over the area yz, we obtain the net power tranmitted along the diretion: a (4) P = P dy + P dy + P dy w = a P + P + P where w i dimenion o waveguide in the z diretion. Power tranmitted in the ubtrate, ilm layer, and ladding i obtained a β oh( γa ψ ) (5) P = ξ R k R[ a ] inh ( R [ γ ] a ) oh ( R [ ψ ]) β R[ γ ] P = ξ R k in( Im[ γ ] a) o( Im[ ψ ]) + Im[ γ ] β oh( γa ψ ) P = ξ R k R[ a ] where ξ = wη, jω / γ =, and µ / η =. erpt rom the Proeeding o the 5 COMSOL Conerene in Boton

4 3. Ue o COMSOL Multiphyi Geometry o the D model i hown in Fig., where ilm i andwihed between a ubtrate and a ladding over. TM polarized wave i guided in the diretion. Thikne o the ilm i mm. Plamoni waveguide are operated at optial or inrared requenie where the metal ha permittivity with negative real part. Dieletri ontant o the layer are given in Table. Figure. Comol model o plamoni waveguide. Table. Dieletri ontant o the layer. Coniguration Subtrate, Film, Cladding, DMD MDM letromagneti wave propagation i governed by Mawell wave equation in requeny domain: µ r jσ ( ) k r = ω (6) guided wave i peretly aborbed by the paive numerial port on the right ide. Thi model onider a etion o a waveguide that i inite in the y diretion. Beaue the ield drop o eponentially outide the waveguide, the ield an be aumed to be zero at ome ditane away rom interae. Thi make the boundary ondition in the y diretion irrelevant, auming that they are impoed uiiently ar away rom interae. Numerial Port boundary ondition require irt olving an eigenvalue problem that olve or the ield and propagation ontant at the boundarie. Study et up onit o two Boundary Mode Analyi tep ollowed by Frequeny Domain tep. Global OD and DA (ge) interae i ued to olve ive tranendental algebrai equation (5), (6), and () in order to ompare numeri and analyti olution. 7. Reult and Diuion Field proile in the DMD waveguide or the ae o k a =. are hown in. The ield etend into the dieletri and metal region, but they are onined to ditane that are le than their ree-pae wavelength. The longitudinal tangential omponent o the eletri ield and tranvere normal omponent o the magneti ield y are ontinuou aro the interae, while longitudinal normal omponent o the eletri ield y i diontinuou. Reult are in agreement with analyti olution hown in Figure 3 by dahed line. letromagneti Wave, Frequeny Domain (emw) phyi interae i ued to olve governing equation (6). letri ield omponent are olved or In-plane vetor. Thi option i appropriated ine there i no outo-plane omponent o the eletri ield or the TM polarization. Numeri Port boundary ondition are applied at the let and right boundarie. Wave eitation i On at the let ide to lunh the guided wave, and O at the right boundarie to avoid bakreletion o the propagating wave, o that erpt rom the Proeeding o the 5 COMSOL Conerene in Boton

5 Figure 3. Field proile in DMD waveguide, k a. =. The diperion urve i hown in Figure 4. Note that propagation ontant i dereaing untion o requeny. For the onidered parameter o the DMD waveguide, TM mode propagate with no uto requeny. A the requeny inreae, the magneti ield tend to be mode onentrated at the metal-dieletri interae, and olution tend to the ingle-interae olution a ω. Again, numerial reult are in agreement with analyti olution hown in Figure 3 by dahed line. Ditribution o the eletri ield or the ae o k a k a =. and =. 7 are hown in Figure 5. Figure 4. Diperion urve or DMD waveguide Reult or the analyi o the MDM waveguide are hown in Figure 6 through Figure 7. Field proile or the ae k a =. 6 are hown in Figure 6. The diperion urve appear in Figure 7. Unlike the DMD waveguide onidered above, the propagation ontant o the MDM waveguide i an inreaing untion o requeny. There i lower uto requeny, whih an be alulate a []: uto + + = 4πa (7) For the MDM waveguide parameter onidered here, numerial value o the uto requeny i uto = 5. 7Tz and uto value o the normalized wavenumber i ( ) =. 54 ka. uto The upper limit or the propagation ontant i deined by the ollowing relation: β β = k + (8) The limit β i the wavenumber o a urae plamon at the metal-dieletri interae a ω. Numerial value o the upper limit i β =.k. Field erpt rom the Proeeding o the 5 COMSOL Conerene in Boton

6 ditribution near the lower uto requeny and upper limit are hown in Figure 5. Figure 5. letri ield ditribution in DMD waveguide at ka =. ( = 9. 54Tz, let) and ka =. 7 ( = 66. 8Tz, right). A an eample o urae plamoni wave propagation in a more ompliated truture, onider a oplanar waveguide. Geometry o the bai urae plamon oplanar waveguide (SP CPW) i hown in Figure 8. A dieletri ubtrate (D) ha metal layer (M) o thikne t patterned on top o the ubtrate. A entral metal layer o width w i eparated by gap ditane g rom the ide wide layer. The modeling methodology i outlined in the veriiation eample o the layered plamoni waveguide. Figure 6. Field proile in MDM waveguide, k a. =.6 Geometry parameter o the analyzed waveguide are: w = g = 5nm, h = nm. Dieletri ubtrate i SiO with dieletri ontant SiO = 3.8. Metal layer are ilver. The Drude model dieletri untion or Ag i [3]: Aγ ( ω) = ω ω ω p ( jγ ) (9) erpt rom the Proeeding o the 5 COMSOL Conerene in Boton

7 5, = 3.8 rad where = 3. 7 ω, p / 3 γ =.736 rad /, whih i onitent with eperimental data by Johnon and Chrity [4]. Operating ree-pae wavelength i λ = 5nm. mode i well onined around the entral metal trip, a hown in Figure. (a) Figure 7. Diperion urve or MDM waveguide. Figure 8. Geometry o urae plamon oplanar waveguide. There are two undamental propagation mode with ubwavelength oninement. Propagation ontant o thee mode are 3, and β / k =.948 j / k =.456 j3.96 β. The irt propagation ontant orrepond to the even mode and the eond repreent the odd mode. Ditribution o the longitudinal omponent o the eletri ield along the enter line o the metal layer or even and odd mode i hown in Figure 9. Power o the propagating (b) Figure 9. Ditribution o the longitudinal eletri ield omponent along the enter line o metal layer or SP CPW (a) even mode and SP CPW (b) odd mode. 8. Conluion Plamoni layered waveguide in DMD and MDM oniguration are onidered and analyzed. Analytial olution provide veriiation o the analyi methodology. Thi veriied modeling tehnique i etended to the analyi o more ompliated geometrial oniguration o urae plamon oplanar waveguide and ubwavelength oninement o the propagating undamental mode i illutrated. 9. Reerene. D. Sarid and W. A. Challener, Modern Introdution to Surae Plamon, Theory, erpt rom the Proeeding o the 5 COMSOL Conerene in Boton

8 Mathematia Modeling and Appliation, Cambridge Univerity Pre, Cambridge,.. letromagneti Wave and Antenna, Sophole J. Oranidi, 3. C. Sönnnihen, Plamon in Metal Nanotruture, Ph. D., Ludwig-Maimilian- Univerity o Munih,. 4. P. B. Johnon and R. W. Chrity, Optial ontant o the noble metal, Phy. Rev. B, vol. 6, no., pp , De. 97. Figure. Power low ditribution or SP CPW even mode (let) and SP CPW odd mode (right). erpt rom the Proeeding o the 5 COMSOL Conerene in Boton