Systematic Design of Antireflection Coating for Semi-infinite One-dimensional Photonic Crystals Using Bloch Wave Expansion

3 Pages·0.16 MB·English

by  Jun Ushida

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
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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

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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.
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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).
In conclusion, we have presented a systematic method
16J.M.BendicksonandJ.P.Dowling,Phys.Rev.E53,4107
for designing a perfect ARC for a semi-infinite 1D PC.
(1996).
We derived exact formulas of structural parameters of 17ThisconditionwasfirstderivedbyS.D.Poissonaccording
the ARC and Fresnel coefficient of the PC. The concept to Ref. [14], p. 2. See also, Z. Knittl, Opt. Acta, 25, 167
of k matching10,11 was not used for our derivations, and
(1978) (in French).
the Bloch wave vector in the infinite PC is not included 18T.Minami,H.Ajiki,andK.Cho,PhysicaE(Amsterdam)
explicitly in Eqs. (2) and (3). The surface immittance 13, 432 (2002).
(admittanceandimpedance)ofthePCplaysanessential 19J. B. Pendry,J. Mod. Opt.41, 209 (1994).
role in the formulas. The adiabatic interconnection8,9,10 20In Ref. [2], p.36 and Ref. [4], p. 21.
can be interpreted to be gradual immittance-matching
structures. Under appropriate approximation, the de-
3