Systematic Design of Antireflection Coating for Semi-infinite One-dimensional Photonic Crystals Using Bloch Wave Expansion Jun Ushida, Masatoshi Tokushima, Masayuki Shirane, and Hirohito Yamada Fundamental Research Laboratories, NEC Corporation, 34 Miyukigaoka, Tsukuba, 305-8501, JAPAN 3 0 We present a systematic method for designing a perfect antireflection coating (ARC) for a semi-infinite 0 one-dimensional(1D)photoniccrystal(PC)withanarbitraryunitcell. WeuseBlochwaveexpansionand 2 time reversal symmetry, which leads exactly to analytic formulas of structural parameters for the ARC n andrenormalizedFresnelcoefficientsofthePC.Surfaceimmittance(admittanceandimpedance)matching a playsanessentialroleindesigningtheARCsof1DPCs,whichisshowntogetherwithapracticalexample. J 5 Revision:4.16 Date:2003−01−0608:54:04+09 ] i c Photonic crystals (PCs) have unique energy disper- s (a) (b) sion due to coupling between periodic materials and l- electromagnetic (EM) waves.1,2,3,4 In particular, the z n(z) z r t strong energy dispersion of the propagation modes of Λ m PCs has attracted much interest because it enables ap- t t at. ppleynisnagtoPrsC,spotloaraizdadt/iodnrofipltmerus,ltaipnldeximeras,gedpisrpoecressisoonrsc.5o,m6,7- Region 3 n3 O R(1eDgi oPnC 3) O m x x These transmission-type applications require a negligi- Region 2 (ARC) d n2 Region 2 (ARC) d n2 - bly small reflection loss at the PC interface.8 There- -d -d d n fore,applyingeffectiveantireflectioninterfacestructures, Region 1 n1 Region 1 n1 o or antireflection coatings (ARCs) to the input and out- 1 r 1 r c put interfaces of the PCs is important.8,9,10,11 Several FIG. 1. Antireflection coating (region 2) with thickness d [ ARCs designed for the PCs have been reported on. The and refractive index n2 for (a) a semi-infinite homogeneous 4 structural ideas of these ARCs were based on the con- medium with refractive index n3 and for (b) a semi-infinite v cepts of adiabatic interconnection8,9,10 or wave vector 1D PC with lattice constant Λ and periodic refractive index 5 matching (k matching)10,11 for two-dimensional PCs. function n(z)(=n(z+Λ)). 0 The reflectance at the interface of one-dimensional (1D) 4 PCs has so far been calculated by plane-wave expan- 0 sion and multiplication of the transfer matrix for a unit 1 cell.12,13,14,15,16 However, these approaches do not tell where k2 is the normal component of the wave vector 2 in the ARC, and r is the reflection coefficient of the directly the optimal ARC parameters, so the numerical i,j 0 light propagating from region i to j.12,13 The reflection / calculation needs to be iterated until these parameters t coefficient in Eq. (1) equals zero for normal incidence a are optimized. In this letter, we derive analytic formulas m of the structural parameters of an ARC that is applied light when d=λ0/4n2 and n2 =√n1n3, where λ0 is the wavelength in vacuum.17 - to a 1D PC.We dealwith 1D systems for simplicity, but d the derivedformulaswouldbe applicable to multidimen- As in Fig. 1 (b), we replace the semi-infinite homoge- n neous medium (region 3) in Fig. 1 (a) by a semi-infinite sional PCs under appropriate approximation. o 1D PC with lattice constant Λ and periodic refractive The structural parameters of a conventional ARC c index function n(z). The function n(z) has an arbitrary : placed between two homogeneous media can be calcu- v spatialmodulation in the unit cell. To determine the re- lated easily if the refractive indices of the two media are Xi known.12,13,14 In Fig. 1 (a), the ARC (region2) with re- flectioncoefficientforthiscase,weexpandtheEMwaves forthesemi-infinitePCbyalltheeigenmodesoftheinfi- fractive index n and thickness d is placed between two r 2 nite PC.18 These eigenmodes, which are extended into a a semi-infinite homogeneous media with refractive indices complex k space (i.e., including the decaying waves), in n (region 1) and n (region 3). These three regions are 1 3 the infinite 1D PC can be obtained as the eigenvectors divided by two boundaries at z =0 and d, where the z − of a transfer matrix15,16,18,19 with a periodic boundary axisisdefinedasperpendiculartothesurfaceoftheARC. condition for a unit cell under a given frequency ω, a Eachregionconsistsoflinearandlosslessdielectrics. The parallel component of a wave vector k , and a polariza- reflection coefficient at z = d is then given by k − tion of light σ. The reflection coefficient r of the electric r +r exp(2ik d) field at the interface (z = d) is determined by the con- r = 1,2 2,3 2 , (1) − tinuityofthetangentialcomponentsofEMwavesattwo 1+r r exp(2ik d) 1,2 2,3 2 1 boundaries (z = 0, d). The expression of the derived − (a) (b) reflection coefficient is formally the same as Eq. (1), ex- 1 0 cept for r2,3. The reflection coefficient r2,3 for normal n=1 n1=2 incidence light is modified into 1 −β2 n)10.5 n)1−0.1 r2,3(ω)= nn22+−NN((ωω)) , (2) αf ( , 0 f = 0 f > −β2 αf ( , −0.2 n=∞ f ≤ −β2 −β2 1 where N(ω) equals Y(±)(ω)/ǫ c, which is a normal- n1=∞ f ≤ −β2 ± k,k 0 −0.5 −0.3 ized surface admittance of the Bloch waves. Note that 0 0.5 1 1.5 2 0 0.5 1 in Y(±)(ω), the “+”(“ ”)sign applies to Y(+) (Y(−)). α α ± k,k − k,k k,k FIG.2. (a) Illustration of sufficient conditions in Eq. (6). ThesurfaceadmittancesY(±) forthetwoorthogonalpo- Theshadedareasindicatetheregionsatisfyingtheconditions larizations are defined as k,k with −β2 =−0.08 (dotted line) as an example. (b) Magnifi- cation of region 0 < α < 1 and −0.3 ≤ f ≤ 0 of Fig. 2(a). Yk(,+k) =Hk,y/Ek,x, Yk(,−k) =Hk,x/Ek,y, at z =+0, (3) Function f with n1 =2 is also plotted. where k is the Bloch wave vector, and Ek,ξ(Hk,ξ) with ξ = x or y stands for the tangential component of the to obtain a positive real number for n , so the explicit electric(magnetic) field of the propagating Bloch waves 2 form of n 1 with Eq.(5) is classified by using positive with a positive group velocity or the decaying Bloch 2 ≥ α Re(N)/n : waves in the positive z direction. These Bloch waves 1 ≡ can be calculated from the forementioned transfer ma- (A) f > β2 for 1<α< trix, hence N(ω) in Eq. (2) is obtained directly. In the − ∞ n 1 (B) f = β2 for α=1 , (6) derivationofEqs. (2) and(3)we usedthe factthat time 2 ≥ ⇔(C) f −β2 for 0<α<1 reversal symmetry20 inhibits the simultaneous appear-  ≤− ance of the propagating and the decaying modes in the same direction for a given set ω, kk and σ in a semi- where f(α,n1)≡(α−1)(α−1/n21), and β ≡Im(N)/n1. { } These sufficient conditions in the rhs of Eq. (6) are infinite 1D PC. Note that the function N(ω) in Eq. (2) illustrated in Fig. 2 (a). As n changes from 1 to , is generally complex due to the phase difference between 1 ∞ the curve for f as a function of α varies from a solid one Ek,ξ and Hk,η (ξ,η = x or y) of the Bloch waves at the to a broken one continuously. The shaded area indicates interface, however the imaginary part of N(ω) is zero if the region satisfying the conditions of Eq. (6). When thesurfaceofthesemi-infinitePCsisamirrorplaneinan 1 < α < , f > β2 always holds because f > 0. The infinite form of the PCs. Note also that multiple reflec- ∞ − case α=1 leads to β =0 due to f =0. A magnification tions in the semi-infinite 1D PCs, which are represented of region 0 <α<1 in Fig. 2 (a) is shown in Fig. 2 (b), as plane waves, are renormalized into a single Fresnel where a curve for function f with n = 2 is added. The coefficientinEq. (2)byusingtheBlochwaveexpansion. 1 thick solid line shows the values of f and α that satisfy The pair values of refractive index n and thickness 2 the third condition in the rhs of Eq. (6) for n = 2 and d of the ARC for semi-infinite 1D PCs that eliminate 1 a given β2. This curve connects two crossing points reflectanceare determinedas follows. The extremalcon- − dition of the reflectance13 R(= r2) with respect to d, (filled circles) of f and −β2 through the minimum point | | (opencircle)off. Thesethree pointsareusedtorewrite i.e., ∂R/∂d=0, is written as the inequality f β2 for 0 < α < 1 in Eq. (6), which ≤ − d 1 r∗ leads to two conditions: (1) the minimum point of f is 2,3 D = ln +m , m=0,1, . (4) less than or equal to β2, and (2) α is located between ≡ λ0/4n2 2πi (cid:20)r2,3(cid:21) ··· the two crossing point−s. WecallD the “normalizedthickness”. Bysubstitutingd Byapplyingtheprevious,wecanobtainthefinalform derivedfrom Eq. (4) into Eq. (1), we obtaina refractive of the three sufficient conditions for n2 1 in Eq. (6): ≥ index n that eliminates the reflectance: 2 (A) 1<α< , (7) ∞ n N 2/Re(N) (B) α=1 and β =0 , (8) 1 n = n Re(N) −| | . (5) 2 1 s n1 Re(N) (C) 0<α<1 , 0 g and − ≤ p √g 1+1/n2 2α √g . (9) The necessary and sufficient condition for the perfect − ≤ 1− ≤ A(N4R)(ωCa)n,fdoarn(ds5e)ωm,.ia-nindfin∂i2tRe/1∂Dd2PC>s0isuDnd≥er0a, ng2iv≥en1ni1n≥Eq1s,. wdihtieorne(gC(β),innc1l)u≡des(cid:0)1n−1 6=1/1n21im(cid:1)2p−lic4itβly2.. CNoontdeitthioantst(hAe)-c(oCn)- dependonn andN(ω)only,sothen neededtoachieve Next, we investigate the physical conditions encapsu- 1 2 ARCs for semi-infinite 1D PCs can be directly obtained lated in Eq. (5). The condition Re(N) > 0 is necessary 2 signing method would to a large extent be applicable (a) (b) (c) to multidimensionalPCsthat havea position-dependent 2 1 n]2 Λ] surface-immittance across the surface. λ/401.5 B1 of c/ 0.8 B5 The authors thank Professor K. Cho for fruitful dis- Thickness d [Units of 0.0151B5 B22BB43 3 4 Frequency [Units 000...0246 0.1 00 BBBB2341 0.50 0.5 1 cfOPAsToiuorhjhrsioneitkssfasiiepo,,wksnuasaKosob,nrr.lKkadicHnS.waKdu.Htazi.IsfiousorNnshkoru,isiah,psekYha,paaro.AomnawrMd.,utiGeTnriYydag.oa.imIuamnkWnsyaopdowttaah,otarJe,Rat,.inOr.TbSayK.w.obnHSuSeoepruriifkregbofaciora(inir,syaofeTalrt,esuCh.PhRieoMtiri,foerouifrMfn.lsdeuas1.idnmsp8Sioaps)airct,oibiutrKoHeostn--..., Refractive Index n Im[k] Re[k] Reflectance 2 [] [Units of 2π/Λ] Funds for Promoting Science and Technology. J. Ushida FIG.3. (a)Pairsof{n2 andd}neededtoachieveARCfor dedicates this work to Professor and Mrs. Cho in honor asemi-infinite1DPC. (b)Dispersion relation ofpropagating of their 60th birthdays. and decaying modes. (c) Reflectance with (solid line) and without (broken line) ARC. by using Eqs. (7)-(9) and (5). Accordingly, thickness d is also determined by Eq. (4) and ∂2R/∂d2 >0. To give a practical example of ARCs for semi-infinite 1E. Yablonovitch,Phys. Rev.Lett. 58, 2059 (1987). 2J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Pho- 1D PCs, in Fig. 3(a) we plot the refractive index and tonic Crystals: Molding the Flow of Light (Princeton Uni- the thickness neededto formthe ARC fora semi-infinite versity Press, Princeton, NJ, 1995). 1D PC. We assume that n =1, that the unit cell of the 1 3K.Ohtaka,Phys. Rev.B 19, 5057 (1979). PC consists of Si (n = 3.5) and SiO (n =1.5) with the 2 4K. Sakoda, Optical Properties of Photonic Crystals same thickness Λ/2, and that the Si layer is the bound- (Springer,Berlin, 2001). ary layer between the semi-infinite PC and the ARC. In 5H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Fig. 3(b),the dispersionrelationofthe infinite 1DPCis Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. plottedandthebandindexisindicatedbyB1-B5. When 74, 1370 (1999); Phys.Rev.B 58, 10096 (1998). the frequency changes along a band, the corresponding 6Y. Ohtera, T. Sato, T. Kawashima, T. Tamamura, and S. pairof[n2(ω)andd(ω)]formsacontinuoustrajectory[as Kawakami, Electron. Lett. 35, 1271 (1999). shown in Fig. 3(a)]. Note that in Fig. 3(a), only condi- 7M. Notomi, Phys.Rev.B 62, 10696 (2000). tion (A) is needed to calculate n2 and d, and frequency 8T. Baba and D. Ohsaki, Jpn. J. Appl. Phys., Part 1 40, regions near the band edges that require n2 4.5 are 5920 (2001). not plotted. Note also that the deviation in th≥e normal- 9Y.Xu,R.K.Lee,andA.Yariv,Opt.Lett.25,755(2000). izedthicknessfrom1isduetothephaseshiftbetweenthe 10A. Mekis and J. D. Joannopoulos, J. Lightwave Technol. electricandthemagneticfieldsattheinterface(z =+0). 19, 861 (2001). InFig. 3(c)weshowthereflectanceofthe1DPCwith 11A.Mekis,S.Fan,andJ.D.Joannopoulos,IEEEMicrowave (solidline)andwithout(brokenline)anARC.TheARC Guid. WaveLett.9, 502 (1999). was designed at a selected frequency (ωΛ/2πc = 0.7), 12P.Yeh,OpticalWavesinLayeredMedia,(Wiley,NewYork, whichisindicatedbyanarrow,whereN(=3.384 i0.499) 1988), pp.86-96. fulfills condition (A). The requiredvalues of n (=−1.868) 13M. Born and E. Wolf, Principles of Optics, 7th ed. (Cam- 2 andD(=1.071)fortheARCarecalculatedfromEqs. (5) bridge UniversityPress, Cambridge, 1999), pp. 63-74. 14H.A.Macleod,Thin-FilmOpticalFilters,3rded.(Institute and (4) with m = 1, which is shown by a circle in Fig. of Physics, Bristol, 2001), Chaps. 2 and 3. 3(a). In Fig. 3(c), the ARC eliminates the reflectance 15P. Yeh, A. Yariv, and C.-S. Hong, J. Opt. Soc. Am. 67, (solid line) at the selected frequency (arrow). 423 (1977). 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