Antireflection Coating at Metamaterial Waveguide Structure by Using Superlattices (LANS)

Transcription

1 Journal of Modrn Physics, 0, 5, Publishd Onlin May 0 in SciRs. Antirflction Coating at Mtamatrial Wavguid Structur by Using Suprlattics (LANS) H. M. Mousa Physics Dpartmnt, Al Azhar Univrsity, Gaza, Palstinian h.mousa@alazhar-gaza.du.ps Rcivd 9 March 0; rvisd 8 April 0; accptd 5 May 0 Copyright 0 by author and Scintific Rsarch Publishing Inc. This wor is licnsd undr th Crativ Commons Attribution Intrnational Licns (CC BY). Abstract Th charactristics of lctromagntic wav rflction and transmission by multilayrd structurs consisting of a pair of lft-handd matrial (LHM) and suprlattics (LANS) slabs insrtd btwn two smi-infinit dilctric mdia ar invstigatd for photovoltaic and solar nrgy applications. Maxwll s quations ar usd to dtrmin th lctric and magntic filds of th prpndicular polarizd wav incidnt at ach layr. Snll s law is applid and th boundary conditions ar imposd at ach layr intrfac to calculat th rflctd and transmittd cofficints of th structur. Th rflctd, transmittd powrs ar dtrmind using ths cofficints by a rcursiv mthod. Th rflctd and transmittd powrs ar computd in both visibl and microwav spctral band with th appropriat LHM for ach band and appropriat location of LANS in th structur. Thy ar illustratd as a function of th incidnt wavlngth, angl of incidnc, magntic fraction of LANS and thicnss of th slabs with th mphasis on th appropriat rfractiv indics. I found that, zro rflctanc and maximum transmittanc of th incidnt powrs ar achivd for visibl spctral band at a singl frquncy if LHM and LANS hav th sam rfractiv indx of opposit signs with th sam width and mor magntic matrial of LANS whil th rflctd powr is zro for lss magntic matrial of LANS in th microwav spctral band which ralizs antirflction coating in this structur. Kywords Angl of Incidnc, Magntic Fraction, Mtamatrial, Microwav Band, Rflction, Transmission, Suprlattics, Visibl Band. Introduction In many applications, rflction is undsirabl and causs insrtion losss, for xampl. It is wll nown that th How to cit this papr: Mousa, H.M. (0) Antirflction Coating at Mtamatrial Wavguid Structur by Using Suprlattics (LANS). Journal of Modrn Physics, 5,

2 application of on or mor antirflction coating (ARC) layr on th front surfac of th photovoltaic clls and optolctronic dvics (Lasrs, IR diods, tc.) rducs th amountd rflction of th incidnt light, which improvs th dvic prformanc []-[3]. Antirflction coatings rquir a particular rfractiv indx and quartr wavlngth thicnss whr it oprats by ovrcoming th mis-match btwn intrinsic impdancs of two mdia. This approach is scalabl ovr a wid spctral rang from microwav to far infrard []. Thr hav bn a fw fforts to dvlop antirflction coatings at THz frquncis by using dilctric mta-matrials or lfthandd matrials (LHM). LHM hav both ngativ prmittivity and prmability and consquntly hav ngativ indx of rfraction [5] [6] and ar abl to match mdia impdancs. Th thory of antirflction coating is xamind by many authors [7] [8]. Th matrix mthod [9] [0] is usually mployd for calculation of rflction cofficint. Chn t al. [] hav prsntd approach of mtamatrial antirflction. It rducs th rflction and nhancs transmission nar a spcifically frquncy ovr a wid rang of incidnc angls for both TE and TM polarizations, Bouhafs t al. [3] hav mad a thortical study of th antirflction coatings on silicon solar clls. Cory t al. [] hav analyzd th rflction and transmission charactristics of a multilayrd structur consisting of mtamatrials and dilctric slabs. In this papr, I invstigat th rflction and transmission proprtis of a suprlattics (LANS)-mtamatrial (LHM)-dilctric multilayrd structur. Sinc antirflction coating is formd by two slabs of th sam thicnss and of opposit rfractiv indics, a pair of LHM and LANS matrials is situatd btwn two smi-infinit dilctric mdia which ar considrd right handd matrials (RHM) of positiv rfractiv indx. Th suprlattic is a latral anti-frromagntic/nonmagntic (LANS) such as f f zn f. LANS ar dscribd with an ffctiv mdium thory. Such dscription is valid whn th wav lngths of th xcitations ar much longr than th suprlattic priod whr L, whr is th magnitud of th wav vctor and L L + L, is th priod of th suprlattic, L and L ar th thicnss of th anti-frromagntic layrs and non-magntic layrs,rspctivly [5] []. f L L and f L L. In th thory th lctric and magntic filds of th incidnt wavs ar dtrmind in ach rgion by Maxwll s quations. Thn Snll s law is applid and th boundary conditions ar imposd at ach intrfac to obtain th rflction and transmission cofficints. Th rflctd and transmittd powrs of th structur ar prsntd in trms of ths cofficints. In th numrical analysis, a rcursiv mthod [3] is usd to calculat th rflctd, transmittd powrs as a function of incidnt wavlngth, angl of incidnc, layr thicnss and magntic fraction of (LANS). Th calculations ar prformd for lctromagntic radiations in both th visibl and microwav bands for a singl wavlngth by slcting th optimum rfractiv indics of LHM and RHM in both bands. Th suitabl LHM in ach band is slctd. Th consrvation law of nrgy is chcd and satisfid.. Thory Considr LHM and LANS of lctric prmittivity and magntic prmability ( ε h, h) and ( ε, ) tivly mbddd btwn two smi-infinit dilctric mdia of prmability and prmittivity as (, ) (, ) rspc- ε and ε. A prpndicular polarizd wav is incidnt on th structur at Y 0 at angl θ rlativ to th normal to th boundary as dpictd in Figur. Figur. Wav propagation through a structur consisting of LHM and LANS matrials insrtd btwn two smi-infinit dilctric mdia. 63

3 Introducing th ffctiv mdium thory, th magntic prmability of th LANS [5] [] which is dscribd as a singl ffctiv mdium, can b writtn as: with whr th xprssions of and ar [5] []: with xx i xy 0 i xy yy ( f f ) ( f f ) f f ( ) xx yy ( f f ), f ( f f ) xy { ( ) ( ) } 0 0 { ( ) ( ) } ω ω ω ω ω + ω ω + ω a m r r ω ω ω ω ω + ω ω + ω a m r r ω πγm, ω γh, ω γh m 0 a a 0 0 ω γ HH + H r a a H a rprsnts an isotropy fild, H th xchang fild, and γ th gyromagntic ratio. m 0 is th sublattic magntization. Th magntic fild of th suprlattic is H 0. Th ffctiv dilctric function of (LANS) is xprssd by [5] [3] as: ε ε f + ε f Th lctric and magntic fild vctors for TE wavs propagating along x-axis with angular frquncy ω ar dfind as: E 0,0, Ez ( ω, y) xp i( lx x ωt) () H Hx ( ω, y), Hy ( ω, y),0 xp i( lx x ωt) Th lctric fild in ach rgion is [], [5]: ily y ily y ( ) E A + B (3) lz l l whr A l and B l ar th amplitud of forward and bacward travlling wavs in th rgion of ordr nlω ( l,,3,), l is th wav vctor insid th matrial and n l is th rfractiv indx of it. c In Suprlattic (LANS) Rgion Th curl Maxwll s quations ar []: By ths quations E 3z y ( 3 3 ) iω H i H o xx x xy y ( ) E ω i H + H 3x 3z o xy 3x yy 3y H i H iωε ε E y 3x 3x 3y o 3z () 635

4 H E E 3z 3x 3x xy 3z iω 0 v y yy x x y y xy whr ν is th Voigt prmability. yy By Maxwll s quation xel iω lhl, th magntic fild in th othr rgions of ordr (,,) H ) ily y ily y ( A B ) + ω lx l ly l ly l l is: Matching th boundary conditions at ach layr intrfac, whr at y 0, Ez Ez and Hx Hx and so on yilds six quations with six unnown paramtrs as: A+ B A + B (5) y ( A B ) ( A B ) (6) y h iy d iy d i3y d i3y d 3 3 A + B A + B (7) iy d i 3 ( ) ( ) y d + i y d i 3 y d A B A3 B3 h v yy y 3y yy 3x xy According to Snll s law 3 with iy ( d+ d3) i3y ( d+ d3) iy ( d+ d3) (9) A B A 3y yy + 3x xy i3y ( d+ d3) i3y ( d+ d3) y iy ( d+ d3) ( A3 B3 ) A v yy, n n ( θ ) x x x x ly ω c l sin () (8) (0) n ε ε yy 3 v, n h h xx For TE polarizd light, at th first intrfac, th Frsnl cofficint (intrfac rflction and transmission (r, t) rspctivly ar givn by [6]: For th othr intrfacs th Frsnl cofficints ar: h y y h y, t hy + y hy + y r G K G K G K r t r t y h y y 3, 3 3, 3 Gy + hk Gy + hk K + Gy K + Gy G K yy 3y + xy 3 x, v yy, y y, 0 whr ( ) Th rflction and transmission cofficints r and t rspctivly of th structur ar [7]: t r B A A A i3y d3 iy d r r r r r r i3y d3 iy d r r + r r + r r t t t 3 3 i ( yd+ 3yd3) i3yd3 iyd r r + r r + r r ( y + 3y 3) i d d ( y + y ) i d 3 d3 ( y + y ) i d 3 d3 () () (3) 636

5 Th rflctanc R and transmittanc T of th structur ar givn by: R rr, T tt y y whr r and t ar th conjugat of r and t rspctivly. Th law of consrvation of nrgy is [7]: () R+ T (5) In this wor, two cass of LHM ar considrd. Th first whn th incidnt lctromagntic wavs in th visibl spctral band and othr on in microwav band. Th frquncy dpndnt prmittivity of LHM in th visibl band is dscribd by Drud mdium modl as [8] whr ω is th angular frquncy, γ is th lctric damping factor. ω p ε( ω) εlattic ω + iωγ For microwav rgion, I mploy a disprsiv LHM with ( ) h (6) ε lattic is th lattic prmittivity, ω p is th ffctiv plasma frquncy and ε h ( ω) Fω p ω ω + iωγ ε ω and ( ) o Fmω mp h ( ω) ω ω + iωγ mo c m ω appard in [5] [9] as: whr ω p and ω mp ar th lctric and magntic plasma frquncis. ω o and ω mo ar th lctric and magntic rsonanc frquncis. F and F m ar th scaling filling paramtrs, γ c and γ m ar th lctric and magntic dissipation factors. 3. Numrical Rsults and Discussion Th paramtrs wr usd in carrying out th numrical calculations ar []: th applid fild H G, 7 m G, H a 00 G, H 50 G, γ.97 0 rad/sc and ε 5.5 for antifrromagntic layrs, ε 8 for th non-magntic layrs. Th rlativ prmability of th dilctrics is 0, th rfractiv indx of th dilctrics n n nd and d d3 d. 3.. In Visibl Spctral Band 6 Th paramtrs wr usd ar [8]: ω p. 0 rad/s, γ. 0 rad/s, ε lattic 9.. Th rlativ prmability of LHM is assumd to b. Th thicnss of ach slab is assumd to b on-half long of th cntral wavlngth. Th rflctd, transmittd powr of th structur is calculatd as a function of wavlngth of th incidnt wavs, angl of incidnc, layr thicnss and magntic fraction of LANS. According to (6) th ral part of rfractiv indx of LHM is ngativ in th wavlngth rang of (500, 600, 700, 000) nm whr th ral part of n of valus (.03,.3, 3.7, to 5.596) whr th damping factor of LHM in this rgion is ignord and no nrgy loss is displayd. Th cntral wavlngth is assumd to b 600 nm. This choic is basd on th spctral stability of th coating and for low rflctanc. Stability mans that th low-rflctanc spctrum changs vry slightly with rfractiv indx variations as shown by Figur. It displays th rflctd, transmittd powr as a function of th normal incidnt wavlngth whn th dilctric rfractiv indx n d changs to th valus of (.3,.86, 6.58). It shows maximum rflctanc R of valu <0.5 and minimum transmittanc T of valu >0.75 at n d of valu.3 ovr a wid wavlngth rang (λ nm). Th rfractiv indics of LHM ar (.3,.86, 6.58) at incidnt wavlngth λ of valus (600, 900, 0) nm rspctivly and that of LANS is (n 3.9 at λ 600 nm and magntic fraction f 0.7 ). n 3 changs vry slightly with frquncy. For ths indics, on minimum appars around λ 600 nm at which th rfractiv indics of LHM, and LANS, dilctrics ar approximatly closd to ach othr and opposit in signs which confirm that high transmittd powr can b achivd if n n, n3 n, d d3 []. Ths conditions which h (7) 637

6 (a) Figur. (a) Th rflctd, (b) transmittd powr as a function of th normal incidnt wavlngth whn th dilctric rfractiv indx n d changs as n d.3,.86, 6.58, f 0.7, d 50 nm. 8 lads to R 0 and T around λ 600 nm ar: r 3 0, t 3, y 3y m, r r3 0.75, t.75, t , tt3 + rr Such AR coating systms ar usd in photodiods (LASER) and othr optolctronic dvics which nd a minimum rflctanc at a singl wavlngth. Figur 3 illustrats th rflctanc spctra ovr a wid rang of incidnc angl for λ of valus (600, 700, 800) nm. In th rang (θ 0, 7 ) th minimum rflctanc is achivd at λ 600 nm. Th implmntation of LANS layr adjacnt to LHM layr dramatically rducs th rflction and gratly nhancs th transmission nar a spcifically frquncy at incidnc angls (θ 0, 7 ). By wavlngth incras to th valu of 800 nm, minimum rflctanc is obsrvd at highr incidnc angl of valu 57. Th ffct of th magntic fraction f on th rflctanc and transmittanc is dscribd in Figur. As f incrass to th valus of (0., 0.5, 0.9), th rfractiv indx of LANS n 3 dcrass to th valus of (.78,.59,.39) whil n of LHM incrass to th valu of (.83 to.86) in th wavlngth rang ( ) nm which lads to rflctanc incras. It is worth to not that, R 0 and T at λ 600 nm, f 0.9. This is bcaus n n.86, n3 n.39, d d3. Figur 5 dmonstrats th ffct of thicnss d of LHM on th rflctanc and transmittanc at normal incidnc of λ 600 nm, 800 nm. Th slab thicnss is changd from 50 nm to 500 nm. It is noticd that rflctanc and transmittanc changs priodically with thicnss. Bsids that, mor transmittanc is ralizd at λ 800 nm whr maximum R is 0.03 whil minimum T 0.97 of th incidnt powr. It is shown that th slctd thicnss (d 50 nm) is appropriat for achiving zro rflctanc at λ 600 nm. 3.. In Microwav Spctral Band For disprsiv LHM with εh ( ω ) and h ( ) (b) ω hav paramtrs appard in [5] as: ω mp π0.95 GHz, ω mo π0. GHz, ω p π3.3 GHz F 0.37, ω o π0.3 GHz, F m 0.6, γc γm 0.. For this LHM, th frquncy rang in which εh ( ω ) and h ( ω ) ar ngativ xtnds from 0. up to.5 GHz with corrsponding wavlngth xtnds from 6 up to 8 mm. Th thicnss of ach of LHM and dilctrics slabs is qual to on half-wavlngth long at th oprating frquncy. Rflctanc and transmittanc ar calculatd numrically as statd abov. Th law of consrvation of nrgy is [0]: R+ T P (8) whr P loss is th loss powr du to losss in LHM. Sinc in th structur arrangmnt shown in Figur, thr is no ffct of magntic fraction of LANS on th rflctd powr and thr is vry small powr loss of LHM which can t b displayd in a figur, I rarrangd th structur to b LANS in Rgion instad of th loss 638

7 Figur 3. Th rflctd powr as a function of th angl of incidnc for diffrnt wavlngth λ 600, 700, 800 nm, n d.86, f 0.7, d 50 nm. (a) (b) Figur. (a) Th rflctd, (b) transmittd powr vrsus th normal incidnt wavlngth whn th magntic fraction f changs as f 0., 0.5,0.9, n d.86, d 50 nm. dilctric which will b in Rgion 3. Th rflction, transmission cofficints ar rwrittn as: r ( ) ( ) h yy y xy x v yy y h yy y xy x v yy y, 639

8 For th othr intrfacs th Frsnl cofficints ar: t h ( yy y + xy x ) ( + ) + h yy y xy x v yy y 3 y h 3y 3 y 3, t3 3y + h3y 3y + h3y r 3y 3 y 3y 3, t3 3y + 3y 3y + 3y r Figur 6 displays th rflctanc, transmittanc and powr loss as a function of th incidnt wavlngth for many valus of magntic fraction. Th oprating wavlngth is assumd to b 0.07 m which is includd in th frquncy rang in which εh ( ω ) and h ( ω ) ar simultanously ngativ. As f incrass to th valus of (0., 0.5, 0.9) n of LANS dcrass to th valus of (.78,.59,.39) rspctivly, n of LHM changs to 9 0 th valus ( i0.7 0 ) to (.37 - i ) in th wavlngth rang of 0.06 to 0.7 m. R 0 and T and zro powr dissipation ar attaind at wavlngth rang λ m and f 0. at θ 30.. Conclusions Th transmission and rflction of prpindicular polarizd wavs by a multilayrd structur consisting of a pair of LHM and LANS matrials mbddd btwn two smi-infinit dilctrics mdia hav bn studid in both visibl and microwav spctral bands with th appropiat LHM and appropiat location of LANS in th h ω of LHM and that of LANS is tan into account. It has bn shown that, th frquncy-dpndnt rfractiv indx of both LHM and LANS plays an important rol in th variation of th rflction cofficints of th structur. Low rflction can b achivd for both visibl and microwav rays by choosing th propr indicis of th matrials constiutd th structur. For incidnt visibl rays, LANS is locatd as shown in Figur R 0 and thn T is attaind at a sigl wavlngth valu of 600 nm with magntic fraction of LANS of valu 0.9, incidnc angl of valu 0 or 7. For incidnt microwav rays, LANS is locatd in Rgion and th dilctric in Rgion 3. R 0 and thn T ar achivd at wavlngth structur. Th frquncy dpndnc of ( ) h ε ω and ( ) (a) Figur 5. (a) Th rflctd, (b) transmittd powr vrsus th layr thicnss d at normal incidnt wavlngth of λ 600 nm, 800 nm, n d.86, f 0.7. (b) 60

9 (a) (b) (c) Figur 6. (a) Th rflctd, (b) transmittd, (c) loss powr vrsus th incidnt wavlngth whn th magntic fraction f changs as f 0., 0.5, 0.9, n d.86, θ 30, d mm, γ γ m 0.. λ 0.06 to 0.7 m with f 0. at θ 30. Th implmntation of LANS adjacnt to LHM layr dramatically rducs th rflction and gratly nhancs th transmission nar a spcifically frquncy. Th law of consrvation of nrgy has bn satisfid by th obtaind rsults. Th obtaind rsults may b usd to rfin th undrstanding of any rlatd applications that may b modld rquiring controlling of rflctd and transmittd powrs as photovoltaic clls and optolctronic dvics. Rfrncs [] Dobrowolsi, J.A., Poitras, D., Ma, P., Vail, H. and Acr, M. (00) Applid Optics,, [] Dinga, A., Valuv, I., Potapin, B. and Lozovi, Y. (0) JOSA A, 8, [3] Bouhafs, D., Moussi, A., Chiouch, A. and Ruiz, M. (998) Solar Enrgy Matrials and Solar Clls, 5, [] Chn, H., Zhou, J., Hara, J., Chn, F., Azad, A.K. and Taylor, A.K. (00) Physical Rviw Lttrs, 05, Articl ID: 6

10 [5] Mousa, H.M. and Shabat, M.M. (0) Applid Physics A,, [6] Mousa, H.M. and Shabat, M.M. (0) Intrnational Journal of Modrn Physics B, 5, [7] Sopori, B.L. and Pryor, R.A. (983) Solar Clls, 8, [8] Rdfild, D. (98) Solar Clls, 3, [9] Bui, D., McCann, M.J., Wbr, K.J. and Dy, C.J. (00) Solar Enrgy Matrials and Solar Clls, 8, [0] Oraizi, H. and Afsahi, M. (009) Progrss in Elctromagntics Rsarch B,, [] Cory, H. and Zach, C. (00) Microwav and Optical Tchnology Lttrs, 0, [] Shabat, M.M. and Mousa, H.M. (007) SPIE Procdings, 658. [3] Vigourux, J.M. (99) Journal of th Optical Socity of Amrica, 8, [] Kong, J.A. (005) Thory of Elctromagntic Wavs. EMW Publishing. [5] Shlby, R.A. (00) Microwav Exprimnts with Lft-Handd Matrials. Ph.D. Thsis, Univrsity of Calfornia, San Digo. [6] Caloz, C. and Itoh, T. (006) Elctromagntic Mtamatrials. John Wily and Sons, Nw Jrsy. [7] Kong, J.A. (00) Progrss in Elctromagntics Rsarch-PIER, 35, [8] Isaac, T.H. (009) Tunabl Plasmonic Structurs for Trahrtz Frquncis. Ph.D. Thsis, Univrsity of Extr, Extr. [9] Pndry, J.B., Holdn, A.J., Swart, W. and Youngs, I. (996) Physical Rviw Lttrs, 76, [0] Cory, H. and Zach, C. (00) Microwav and Optical Tchnology Lttrs, 0,