1. INTRODUCTIONThe scattering of electromagnetic waves by multilayeredperiodic structures continues to be an active and vibrantresearch topic, which is stimulated by the increasing uti-lization of grating configurations in optoelectronic de-vices, photonic bandgap crystals, antennas, and othertechnological applications. In this context, solutions forgrid or crossed surface-relief gratings were initiated morethan 20 years ago,1–4 and later studies5–20 of such two-dimensional (2D) gratings in three-dimensional (3D) ge-ometries have developed a variety of field formulations,which may broadly be classified as differential, coupled-wave, or modal representations. However, the efficientnumerical modeling of 3D grating structures remains achallenging problem for which new alternative formula-tions may provide greater physical insights or offer nu-merical advantages. With such possibilities in mind, wepresent here a recently developed modal approach21–23
using expansions of both the lattice (space) vectors andthe reciprocal lattice (wave number) vectors in each grat-ing layer. Together with a transmission-line formulationof the boundary-value problem at the layer interfaces,this approach can systematically be applied to explorewave scattering in a wide class of 3D periodic structures.
The configurations considered here consist of dielectric
materials inside layers stacked along the vertical z axis.These materials can be absorbing, and each layer mayhave a periodic form along two directions in the horizon-tal xy plane, as shown in Fig. 1. Furthermore, any of themedia can generally be characterized by a diagonal per-mittivity tensor; i.e., biaxial properties can be accommo-dated if they are oriented along the principal x, y, and zaxes. Any number of layers such as that in Fig. 1 mayoccur along the z direction, but all periodic layers must becommensurate; i.e., a common periodic cell can be definedin the xy plane. We can thus treat 3D structures havinga 2D horizontal periodicity, and the vertical distributionof the layers is assumed to be arbitrary so that it may beperiodic along z. Hence our approach can also addressstructures characterized by a full 3D periodicity, as foundin photonic bandgap crystals. We are mostly interestedhere in situations that provide strong scattering or guid-ing of electromagnetic waves. This property implies thatthe periodic layers may exhibit large variations in theirdielectric constants, and the length scale of those varia-tions is comparable to the lattice spacing, which is itselfon the order of the wavelength.
Consistent with Fig. 1, our present work applies to ar-bitrary forms of lattice structures as well as to arbitrarydielectric variations within the unit cell. In our ap-
2002 Optical Society of America
2006 J. Opt. Soc. Am. A/Vol. 19, No. 10 /October 2002 Lin et al.
proach, the structure within a unit cell in every layer iscompletely separated from the underlying lattice configu-ration, which is specified by defining primitive lattice vec-tors. Many shapes of the components inside the unit cellcan then be treated without requiring staircase or otherapproximations. By using lattice vectors together withtheir associated reciprocal lattice vectors, our field repre-sentation can thus handle arbitrary lattice configurationsin a natural fashion. After obtaining the characteristic(modal) fields inside a lattice of each periodic layer, weconstruct the complete solution for grating structuresthat contain an arbitrary number of such layers by usingequivalent transmission-line concepts. Such solutionshave already been shown24,25 to be very effective in thecontext of 2D geometries involving linear (one-dimensional) periodicities. The present paper thus ex-tends transmission-line representations to wave scatter-ing by gratings in 3D configurations.
We recognized that previous work has also treatedgratings in 3D geometries involving anisotropicmedia,17,18,20 and gratings having arbitrary profiles orcomplex (e.g., nonrectangular) lattice shapes.8–10,12,14–20
The expansions presented here for both the lattice vectorsand the reciprocal lattice vectors can accommodate thesefeatures in a natural and relatively simple manner. Inthe context of wave scattering by 3D gratings and to thebest of our knowledge, such an approach has not beenpreviously reported to the optics community. The mainadvantages of this approach are its coordinate-free formu-lation and very compact notation, which can readily begeneralized to arbitrary dimensions by simply redefiningthe real-space and reciprocal lattice vectors.
To appreciate the role of the real and reciprocal latticeexpansions, we outline in Section 2 certain methods andpertinent results commonly used in condensed-matterphysics for treating general higher-dimensional periodicmedia. The transverse components of the electromag-netic fields are then derived from Maxwell’s equations inSection 3 for an inhomogeneous layer having biaxial prop-erties along the principal axes. Field expressions for ahomogeneous isotropic layer and a homogeneous uniaxiallayer are then obtained in Subsections 3.A and 3.B, re-spectively. Subsection 3.C deals with the expressions forthe fields in a biaxial periodic layer. Transmission-linemethods are used in Section 4 to address the requiredboundary-value problem imposed by the presence of layer
Fig. 1. Two-dimensional periodic variation of the dielectric con-stant within the xy plane. The lattice can be specified by arbi-trary lattice vectors a1 and a2 .
interfaces. We illustrate the flexibility of our approach inSection 5 by comparing its numerical results with thoseobtained by others and by applying it to a practical butcomplex situation involving many layers and an absorb-ing uniaxial region. Issues and remarks that provide abetter perspective on some of the results are relegated toseveral appendixes.
2. LATTICE STRUCTURE AND ITSRECIPROCAL WAVE-VECTOR FORMTo treat periodic layers such as those shown in Fig. 1, thedielectric constant e in any of the horizontal layers is as-sumed to be independent of z but can otherwise be de-scribed by an arbitrary periodic function within the xyplane. When the position vector r is written in the formr 5 r 1 zz, where r lies in the xy plane, e then becomes afunction of r only. We further assume that the periodici-ties of the layers are commensurate with each other sothat a common 2D unit cell with primitive lattice vectorsa1 and a2 in the xy plane can be identified for all the lay-ers. The lattice vectors R in the xy plane are then givenby R 5 m1a1 1 m2a2 , where m1 and m2 are arbitraryintegers. Within any periodic layer, the dielectric con-stant must have the same periodicity as the lattice; i.e.,for any lattice vector R we have
e~r 1 R! 5 e~r!. (1)
If the lattice is rectangular, the primitive lattice vectorsa1 and a2 are perpendicular to each other. However, thepresent treatment can accommodate arbitrary primitivelattice vectors, thus allowing for different lattice configu-rations.
To deal with such general (nonrectangular) lattices, weconsider the reciprocal lattice associated with the period-icity. Primitive lattice vectors for the reciprocal latticeare then defined in terms of the primitive lattice vectorsby b1 5 2pa2 3 z/V and b2 5 2pz 3 a1 /V, where V5 u(a1 3 a2) • zu is the size of a unit cell. The relation-ships among the primitive lattice vectors and the primi-tive reciprocal lattice vectors are
al • bj 5 2pd l, j for l, j 5 1 or 2, (2)
where d l, j is the Kronecker delta function. The recipro-cal lattice vectors are then given by K 5 n1b1 1 n2b2 ,where n1 and n2 are arbitrary integers. Note that
exp~iK • R! 5 1 (3)
must be satisfied for arbitrary choices of R and K.Any function f(r) having the same periodicity as the
lattice can be expanded as a Fourier series:
f~r! 5 (K
fK exp~iK • r!, (4)
where the sum over K implies a double sum over integers,such as n1 and n2 , that specify a given K, and the Fouriercoefficients fK are given by
fK 5 Ecell
dr
Vf~r!exp~2iK • r!, (5)
where the integration extends over a unit cell.
Lin et al. Vol. 19, No. 10 /October 2002 /J. Opt. Soc. Am. A 2007
The following two exact relations are then very usefulin dealing with general periodic systems in higherdimensions26:
Ecell
dr
Vexp(i~K 2 K8! • r) 5 dK,K8 (6)
(K
exp(iK • ~r 2 r8!) 5 V(R
d ~r 2 r8 1 R!, (7)
where d (r 2 r8 1 R) is the multidimensional Dirac deltafunction. Here the summation over R is actually adouble sum over integers, such as m1 and m2 , thatspecify a given R. If K 5 n1b1 1 n2b2 and K8 5 n18b11 n28b2 then dK,K8 5 dn1 ,n18
dn2 ,n28.
In the case that f(r) has a constant value fa inside acertain ‘‘atomic’’ region and a constant value fb in the sur-rounding ‘‘background’’ region, it is convenient to expressf(r) in Eq. (5) as
f~r! 5 fb 2 fbF1 2f~r!
fbG . (8)
The first term is constant within the unit cell, and there-fore, according to Eq. (6), it yields a Dirac delta function.The second term has no contribution within the back-ground region, and therefore we need to integrate onlyover the region occupied by the ‘‘atom.’’ Hence the Fouriercoefficient can be expressed in the form
fK 5 fbdK,0 1 ~ fa 2 fb!Eatom
dr
Vexp~2iK • r!. (9)
In Section 5 we will present results for rectangular, circu-lar, and triangular atoms. Exact analytical expressionsfor the Fourier coefficients for the dielectric constants andtheir inverses can then be easily obtained by using theabove formula.
To relate the above to the scattered fields, we let kt be atransverse wave vector in the xy plane. This vector is aprescribed quantity in the case of scattering problems,but it is part of the solution in the case of guided waves.As customary, we shall restrict kt to lie within the firstBrillouin zone of the reciprocal lattice. Recalling thatany wave vector in the xy plane can be obtained by addingto kt an appropriate reciprocal lattice vector K, we findthat the most general wave vector is given by
kK 5 kt 1 K. (10)
According to Bloch’s theorem, any component of the elec-tric and magnetic fields discussed below must then varywith r in the form exp(ikK • r)uK(r), where uK(r) is a pe-riodic function having the same periodicity as the real lat-tice.
3. MODAL FIELDS IN INHOMOGENEOUSBIAXIAL LAYERSFor time-harmonic fields within an inhomogeneous di-electric medium characterized by a biaxial (diagonal) ten-sor with respect to the principal axes, Maxwell’s equa-tions take the form
where a time variation exp(2ivt) is assumed but sup-pressed. We first derive equations obeyed by the trans-verse (x and y) components of the electric and magneticfields for biaxial terms ex , ey , and ez , all of which are as-sumed to be independent of z. These equations are thenused to derive expressions for the transverse fields in ho-mogeneous as well as periodic layers.
Expressing the longitudinal (z) field components interms of the transverse ones by using Eq. (13), we get
Ez 5i
ve0
1
ez~]xHy 2 ]yHx!,
Hz 52i
vm0~]xEy 2 ]yEx!. (14)
Using these expressions to eliminate the z components,we find that the transverse ones obey
]zS Ex
EyD 5 iF 2
1
ve0]x
1
ez]y vm0 1
1
ve0]x
1
ez]x
2vm0 21
ve0]y
1
ez]y
1
ve0]y
1
ez]x
G3S Hx
HyD , (15)
]zS Hx
HyD 5 iF 1
vm0]x]y 2ve0ey 2
1
vm0]x
2
ve0ex 11
vm0]y
2 21
vm0]x]y
G S Ex
EyD .
(16)
In the above equations the transverse components ofthe electric and magnetic fields are still coupled to eachother. They can be decoupled by differentiating Eqs. (15)and (16) with respect to z. The transverse magneticfields can then be eliminated to obtain the exclusivelytransverse form:
]z2S Ex
EyD
5 F 2k02ex 2 ]x
1
ez]xex 2 ]y
2 ]x]y 2 ]x
1
ez]yey
]x]y 2 ]y
1
ez]xex 2k0
2ey 2 ]x2 2 ]y
1
ez]yey
G3 S Ex
EyD . (17)
However, we see that the x and y components of theelectric fields are still generally coupled to each other. Asa result, the modes of a general inhomogeneous layer can-not be separated into pure transverse electric and trans-
2008 J. Opt. Soc. Am. A/Vol. 19, No. 10 /October 2002 Lin et al.
verse magnetic types, but they are of mixed (hybrid) type.Alternatively, we can eliminate the transverse electricfields to similarly obtain a single equation involving onlythe transverse magnetic fields:
]z2S Hx
HyD
5 F 2k02ey 2 ]x
2 2 ey]y
1
ez]y 2]x]y 1 ey]y
1
ez]x
2]x]y 1 ex]x
1
ez]y 2k0
2ex 2 ex]x
1
ez]x 2 ]y
2G3 S Hx
HyD . (18)
A. Homogeneous Isotropic LayersFor a homogeneous isotropic layer where ex 5 ey 5 ez5 e is a constant, Eq. (17) yields
]z2S Ex
EyD 5 F2k0
2e 2 ]x2 2 ]y
2 0
0 2k02e 2 ]x
2 2 ]y2G S Ex
EyD .
(19)
Since the fields within a periodic layer must obey Bloch’stheorem, the fields within the homogeneous layers mustdo so as well. We therefore substitute
S Ex
EyD 5 exp~ikKz !exp~ikK • r!S Ex0
Ey0D (20)
into Eq. (19) to obtain an eigenvalue problem. Its eigen-values turn out to be doubly degenerate and are given by
kK2 5 k0
2e 2 kK2 , (21)
where we have defined
kK 5 @~kK!x2 1 ~kK!y
2#1/2, (22)
and the vector kK was defined in Eq. (10). In the case ofscattering in a lossless medium, we let
kK 5 HAk02e 2 kK
2 if k02e . kK
2
iAkK2 2 k0
2e if k02e , kK
2, (23)
where the square-root signs are defined25 by an appropri-ate branch cut if absorbing media are involved. Theeigenvectors corresponding to these eigenvalues can bechosen as
S 10 D , S 0
1 D . (24)
Notice that this is not the only choice of eigenvectors, be-cause any two linearly independent vectors can be used.However, this particular choice is certainly the simplestand has been used in previous studies.8,11 The most gen-eral solution for the electric field can then be written as
S Ex~r!
Ey~r! D 5 (K
exp~ikK • r!H S fK,x
fK,yD
3 exp~ikKz ! 1 S gK,x
gK,yD exp@ikK~t 2 z !#J , (25)
where, within a given layer, we have adopted a local coor-dinate along z such that 0 < z < t with t being the layerthickness. From the electric field, Eq. (25) yields themagnetic field
S Hy~r!
2Hx~r! D 5 Ae0
m0(K
exp~ikK • r!YKH S fK,x
fK,yD
3 exp~ikKz ! 2 S gK,x
gK,yD exp@ikK~t 2 z !#J ,
(26)
where
YK 5 F ~kK!x2 1 kK
2
k0kK
~kK!x~kK!y
k0kK
~kK!x~kK!y
k0kK
~kK!y2 1 kK
2
k0kK
G (27)
can be interpreted as the generalized characteristic ad-mittance of the medium. Notice that in expressing themagnetic field vector in Eq. (26), the y component wastaken before the x component so that the equations de-rived later on become more symmetrical. We have alsoinserted a minus sign in the x component in order to makethe admittances obtained later conform to their commontransmission-line definition25; i.e., YK reduces to the cor-responding well-known results for the special case wherethe system is invariant with respect to x or y, as shownbelow in Eq. (28).
Note in Eq. (23) that plane waves whose K satisfiesk0
2e . kK2 represent waves propagating along the z direc-
tion. On the other hand, plane waves whose K satisfiesk0
2e , kK2 refer to waves that are evanescent along the z
direction. In most applications, the frequencies of inter-est are typically low enough so that only a few propagat-ing waves occur. Because evanescent waves often con-tribute little to the physical quantities of interest, thepresent method is computationally very efficient for mostpractical purposes.
To compare the above results with results for lower-dimensional periodic structures, it is interesting to con-sider the special case in which the medium is invariantalong y but is periodic along x. We furthermore assumethat the transverse wave vector kt is along the x direction,in which case we can set the y component of the vectorskK equal to zero. It is easy to see that YK then reduces to
YK 5 F k0e
kK
0
0kK
k0
G . (28)
For the field expressions, we can set fK,x and gK,x 5 0,which means that Ex 5 Hy 5 0. From Eq. (14) we seethat Ez also vanishes so that the electric field is confinedto the y direction and the wave represents a TE modewith YK 5 kK /k0 . On the other hand, if we set fK,y5 gK,y 5 0, then Hx 5 Ey 5 Hz 5 0. The magnetic
Lin et al. Vol. 19, No. 10 /October 2002 /J. Opt. Soc. Am. A 2009
field then points in the y direction and the wave repre-sents a transverse magnetic wave with YK 5 k0e/kK . Inboth cases, one recovers expressions for the fields and ad-mittances that are identical to those in 2D geometries.25
B. Homogeneous Uniaxial LayersIn a homogeneous uniaxial layer we have ex 5 ey 5 e i
and ez 5 e' , with both e i and e' being constants. Forsuch a layer, Eq. (17) becomes
]z2S Ex
EyD 5 F 2k0
2e i 2e i
e'
]x2 2 ]y
2 S 1 2e i
]zD ]x]y
S 1 2e i
]zD ]x]y 2k0
2e i 2 ]x2 2
e i
]z]y
2G3S Ex
EyD . (29)
Expressing the electric field as in Eq. (20) and substi-tuting it into Eq. (29), we obtain the following eigenvalueequation:
One eigenvalue solution is given by
kK,12 5 k0
2e i 2e i
e'
kK2 , (31)
with a corresponding normalized eigenvector
S Ex,0
Ey,0D 5
1
kKS ~kK!x
~kK!yD . (32)
The other eigenvalue solution is given by
kK,22 5 k0
2e i 2 kK2 , (33)
and its corresponding normalized eigenvector is
S Ex,0
Ey,0D 5
1
kKS 2~kK!y
~kK!xD . (34)
The most general solution is therefore given by
S Ex
EyD 5 (
Kexp~ikK • r!AK
3 S fK,x exp~ikK,1z ! 1 gK,x exp@ikK,1~t 2 z !#fK,y exp~ikK,2z ! 1 gK,y exp@ikK,2~t 2 z !# D ,
(35)
with
AK 51
kKF ~kK!x 2~kK!y
~kK!y ~kK!xG . (36)
The magnetic field can be obtained by using Eq. (35) inEq. (16), which yields
S Hy
2HxD 5 Ae0
m0(K
exp~ikK • r!AKYK
3 S fK,x exp~ikK,1z ! 2 gK,x exp@ikK,1~t 2 z !#fK,y exp~ikK,2z ! 2 gK,y exp@ikK,2~t 2 z !# D ,
(37)
where
YK 5 F k0e i
kK,1
0
0kK,2
k0
G . (38)
The matrix YK can be interpreted as the generalized char-acteristic admittance of the homogeneous uniaxial me-dium.
The case of scattering at normal incidence requires spe-cial attention because then kt is zero and therefore theK 5 0 elements in (kK)x and (kK)y are both identically
zero. Consequently the two eigenvectors given above inEqs. (32) and (34) are not defined. The above expressionsfor the fields therefore have incorrect K 5 0 terms.However, their corresponding eigenvalues become degen-erate and are correctly given by Eqs. (31) and (33) ask2 5 k0
2e i . Any choice of eigenvector will satisfy the ei-genvalue equation in Eq. (30). The problem can then beeasily resolved by arbitrarily choosing any two normal-ized and mutually orthogonal eigenvectors for the K 5 0components. For example, we can use
S Ex,0
Ey,0D 5 S 1
0 D , S 01 D . (39)
A very simple way to deal numerically with the problem isto perturb kt away from zero by assigning it a very smallmagnitude. The computed numerical results should beinsensitive to the direction given to kt .
In Appendix A we will see how the results derived herefor homogeneous uniaxial layers reduce to those derivedin Subsection 3.B for homogeneous isotropic layers.
C. Periodic LayersExpressions for the waves within a periodic layer havinga rectangular lattice associated with two mutually per-pendicular gratings were derived by Peng and Morris,11
who used field expansions based on modes defined by aneigenvalue problem. Those results are extended here tocover arbitrary lattice structures.
F k2 2 k02e i 1
e i
e'
~kK!x2 1 ~kK!y
2 2S 1 2e i
ezD ~kK!x~kK!y
2S 1 2e i
ezD ~kK!x~kK!y k2 2 k0
2e i 1 ~kK!x2 1
e i
ez~kK!y
2G S Ex,0
Ey,0D 5 0. (30)
2010 J. Opt. Soc. Am. A/Vol. 19, No. 10 /October 2002 Lin et al.
For this purpose, we convert the differential forms ofEqs. (17) and (18) into algebraic ones by expressing thetransverse fields as
S Ex
EyD 5 (
Kexp~ikK • r!(
mS ~PK,m!x
~PK,m!yD
3 $ fm exp~ikmz ! 1 gm exp@ikm~t 2 z !#%,
(40)
S Hy
2HxD 5 Ae0
m0(K
exp~ikK • r!(m
S ~QK,m!x
~QK,m!yD
3 $ fm exp~ikmz ! 2 gm exp@ikm~t 2 z !#%.
(41)
As was the case for Eq. (26), we have reversed the sign ofHx and interchanged the position of the two transversemagnetic components in Eq. (41). In the modal method,the fields are expressed in terms of a sum of the modes ofthe periodic layer, each of which is labeled by the modeindex m. These modes are determined by solving an al-gebraic system of equations, which is derived below.
In Eqs. (11)–(18), each biaxial component of e has thesame periodicity as the lattice, so that it can be expandedas a Fourier series:
ea~r! 5 (K
eK,a exp~iK • r!, (42)
where a 5 x, y, or z and the Fourier coefficients eK,a aregiven by
eK,a 5 Ecell
dr
Vea~r!exp~2iK • r!. (43)
The inverse of e(r) can also be expanded as
ga~r! 51
ea~r!5 (
KgK,a exp~iK • r!, (44)
where its Fourier coefficients are given by
gK,a 5 Ecell
dr
V
1
ea~r!exp~2iK • r!. (45)
Clearly the Fourier coefficients for ga(r) can be obtained,in principle, from those for ea(r) by inverting the infinite-dimensional matrix obtained from the Fourier coefficientsof ea(r). However, because the Fourier series must all betruncated to a finite size, the exact relationship betweenthe Fourier coefficients of ga(r) and ea(r) holds only ap-proximately. This has a significant effect on the accuracyand convergence rates of the matrix truncations involvedin computing the various field quantities. In this con-text, Li has developed14,20,27 criteria that ensure rapidconvergence. However, for lattices and components hav-ing other than parallelogram shapes, these criteria re-quire that those other shapes be approximated in stair-case form by subdividing them into many parallelograms.Because our approach does not require such a subdivi-sion, we have adopted other procedures to improve theconvergence process, as discussed in Section 5.
To convert Eqs. (15) and (16) into algebraic forms, wemultiply Eqs. (40) and (41) by exp(2ikK • r), integrate rover a unit cell, and apply Eq. (6). The first component ofEq. (15) then yields
(m
km~PK,m!x$ fm exp@ikmz# 2 gm exp@ikm~t 2 z !#%
5 k0(m
~QK,m!x$ fm exp~ikmz ! 2 gm exp@ikm~t 2 z !#%
2~kK!x
k0(K8
gK2K8,z(m
@~kK8!y~QK8,m!y
1 ~kK8!x~QK8,m!x#
3 $ fm exp~ikmz ! 2 gm exp@ikm~t 2 z !#%. (46)
As Eq. (46) holds for any z inside the periodic layer, weobtain
km~PK,m!x 5 k0~QK,m!x 2~kK!x
k0(K8
gK2K8,z
3 @~kK8!y~QK8,m!y 1 ~kK8!x~QK8,m!x#,
(47)
which can be rewritten as
(K8
dK,K8
km
k0~PK8,m!x
5 (K8
FdK,K8 2~kK!x
k0gK2K8,z
~kK8!x
k0G ~QK8,m!x
2~kK!x
k0(K8
gK2K8,z
~kK8!y
k0~QK8,m!y .
(48)
The remaining three equations from Eqs. (15) and (16)can similarly be treated to get
(K8
dK,K8
km
k0~PK8,m!y
5 (K8
FdK,K8 2~kK!y
k0gK2K8,z
~kK8!y
k0G ~QK8,m!y
2~kK!y
k0(K8
gK2K8,z
~kK8!x
k0~QK8,m!x , (49)
(K8
dK,K8
km
k0~QK8,m!x
5 (K8
F eK2K8,y 2 dK,K8
~kK8!x2
k02 G ~PK8,m!y
1 (K8
~kK8!x
k0
~kK8!y
k0dK,K8~PK8,m!x , (50)
Lin et al. Vol. 19, No. 10 /October 2002 /J. Opt. Soc. Am. A 2011
(K8
dK,K8
km
k0~QK8,m!y
5 (K8
F eK2K8,x 2 dK,K8
~kK8!y2
k02 G ~PK8,m!x
1 (K8
~kK8!x
k0
~kK8!y
k0dK,K8~PK8,m!y . (51)
It is convenient to combine and simplify the form ofthese equations by adopting matrix notation and definethe vectors
P 5 S Px
PyD , Q 5 S Qx
QyD , (52)
and 2 3 2 block matrices given by
U 5 F I 2 xgzx 2xgzy
2ygzx I 2 ygzyG , (53)
V 5 F ex 2 y2 yx
xy ey 2 x2G . (54)
In the above equations, I is the identity matrix, and x andy are diagonal matrices given respectively by dK,K8(kK)xand dK,K8(kK)y . The matrix ea has elements eK2K8,a ,and the matrix gz has elements gK2K8,z . Notice that Uand V are symmetric matrices. For convenience, we nor-malize distances in units of l, where l is the vacuumwavelength, and also normalize all wave vectors, recipro-cal lattice vectors, and km in units of k0 5 2p/l.
In terms of these quantities, Eqs. (48)–(51) are ex-pressed as
U Q 5 kP, (55)
V P 5 kQ, (56)
from which Q can be eliminated to get
U V P 5 k2P. (57)
Hence k2 are the eigenvalues of the matrix U V, and Pare the corresponding eigenvectors. Once the eigenval-ues and eigenvectors have been determined, the vector Qis given by
Q 5 V Pk21 or Q 5 U21 Pk. (58)
Notice that Eq. (57) can also be derived more directlyfrom Eq. (17).
Alternatively, we can eliminate P from Eqs. (55) and(56) to get
V U Q 5 k2Q. (59)
Hence k2 are also the eigenvalues of the matrix V U, andQ are the corresponding eigenvectors. The vector P canthen be found from
P 5 V21 Qk or P 5 U Qk21, (60)
which can also be derived directly from Eq. (18).Notice that Eqs. (57) and (59) yield exactly the same set
of eigenvalues k2 because U V and V U are transpose ma-trices. The matrix U V is a 2 3 2 block matrix given by
L 5 U V 5 F ex 2 xgzxex 2 y2 yx 2 xgzyey
xy 2 ygzxex ey 2 ygzyey 2 x2G .
(61)In the special case of a rectangular lattice, these equa-tions reduce to those first derived by Peng and Morris.11
The quantities km , (PK,m)x , (PK,m)y , (QK,m)x , and(QK,m)y are determined solely by the underlying latticeand the structure of the unit cell of the periodic layer un-der consideration; i.e., they are independent of the param-eters of any other layers. In contrast, the modal coeffi-cients fm and gm in Eqs. (40) and (41) are not directlyrelated to the underlying structure of the periodic layerbut rather are determined from the required boundaryconditions. Specifically, those coefficients depend on thethickness of the layer and the nature of the other layersstacked above and below it. This problem is then prefer-ably treated by a transmission-line approach, as dis-cussed below.
4. TRANSMISSION-LINE TREATMENT OFBOUNDARY CONDITIONSFrom Eqs. (40) and (41) for the transverse electric andmagnetic fields, one can employ well known transmission-line methods to solve the required boundary-value prob-lem for the entire structure. Both the scattering charac-teristics and the guided wave properties can then beobtained in a systematic fashion.
For this purpose we rewrite these expressions for thejth layer as
S Ex~r!
Ey~r! D ~ j !
5 (m
em~ j !~r!vm
~ j !~z !, (62)
S Hy~r!
2Hx~r! D ~ j !
5 (m
hm~ j !~r!im
~ j !~z !, (63)
where
em~ j !~r! 5 (
Kexp~ikK • r!S ~PK,m!x
~PK,m!yD ~ j !
(64)
hm~ j !~r! 5 Ae0
m0(K
exp~ikK • r!S ~QK ,m !x
~QK,m!yD ~ j !
(65)
are the modal transverse electric and magnetic fields, and
vm~ j !~z ! 5 fm
~ j ! exp~ikm~ j !z ! 1 gm
~ j ! exp@ikm~ j !~t ~ j ! 2 z !#,
(66)
im~ j !~z ! 5 fm
~ j ! exp~ikm~ j !z ! 2 gm
~ j ! exp@ikm~ j !~t ~ j ! 2 z !#
(67)
are modal voltage and current amplitudes, respectively.We have added a superscript j to indicate that the variousquantities belong to a specific ( jth) layer. Imposing therequired boundary condition between the jth and ( j1 1)th layers, we have
(m
S ~PK,m!x
~PK,m!yD ~ j !
vm~ j !~t ~ j !! 5 (
mS ~PK,m!x
~PK,m!yD ~ j11 !
vm~ j11 !~0 !,
(68)
and
2012 J. Opt. Soc. Am. A/Vol. 19, No. 10 /October 2002 Lin et al.
(m
S ~QK,m!x
~QK,m!yD ~ j !
im~ j !~t ~ j !! 5 (
mS ~QK,m!x
~QK,m!yD ~ j11 !
im~ j11 !~0 !.
(69)
Applying Eqs. (68) and (69), we can now use thetransmission-line formulation25,28 to determine all vm
( j)
and im( j) , and therefore all fm
( j) and gm( j) , associated with
any combination of homogeneous and periodic layers.Both scattering and waveguiding problems can then behandled systematically. This approach not only facili-tates physical insights into the problem but, as discussedin Section 5, provides important numerical advantages.
5. NUMERICAL CONSIDERATIONS ANDEXAMPLESAs is necessary for solutions involving infinite matrices,truncations of the order to N 3 N must be used in calcu-lating any of the field components. The numerical con-vergence of the results as N increases is then an impor-tant problem, which has been explored in detail by Li14,20
and others. In particular, Li has developed criteria thatensure rapid convergence for the relevant truncations.Basically, these criteria specify that either Eq. (42) for e(r)or Eq. (44) for g (r) must be used to express the variationof the dielectric constant (and its inverse) inside the unitcell. Li’s criteria can be readily implemented in 2Dgeometries,28 but 3D situations having lattices and ele-ments whose shapes are not parallelograms have beenhandled by staircase approximations14,20 or by transfor-mations of the fields into normal and tangential compo-nents with respect to the media boundaries.18 Becauseour approach avoids such procedures, we have not di-rectly applied Li’s criteria14,20 but have used other strat-egies to improve convergence, as discussed below.
Specifically, we have found that for geometries of thetype shown in Fig. 1, our approach can achieve rapid con-vergence even at high dielectric contrast that may reachvalues of 10 or more, as illustrated below for a quantum-well configuration. This can be achieved by a judiciouschoice of the Fourier expansions describing the structureof the unit cell. In general, those expansions involvethree separate (x, y, and z) components each. As an ex-ample, we could choose representations using a set(ex , ey , gz), which implies that only the matrices for ex ,ey , and gz or their inverse matrices are used. One maythus alternatively employ other combinations of e and/org, for a total of 16 choices. By carrying out extensive cal-culations, we have found that all these choices convergereasonably well for low dielectric contrasts, i.e., if theatomic (cylinder) and background (connected) regions
have dielectric constants whose ratio is close to 2. Forlarger dielectric contrasts, however, it appears that thebest choice is (gx , gy , ez) if the dielectric constant in theatomic region is denser, or (ex , ey , ez) if the backgrounddielectric constant is larger.
To verify the validity of our approach, we applied it tosituations for which numerical results have been re-ported. In particular, we reproduced Figs. 18 and 19given by Grann et al.9 and Fig. 2 by Peng and Morris.11
With the same choice for the e(r) and g (r) expansionsthat they used, and to the accuracy that we could distin-guish in their published curves, our results were identicalto theirs. However, our transmission-line representationreduces the size of all relevant matrices to one half com-pared with the results of their algebraic method.
In addition, we applied our approach to calculate theresults given in Table 2 of the paper by Li.14 This situa-tion involves circular cylinders, for which Li used a 20003 2000 staircase grid approximation in conjunction withappropriate handling of the Fourier expansions of the di-electric constants. Using our choice of the (gx , gy , ez)set as mentioned above, we compare the results in Table1, and the agreement is seen to be very good. The num-ber of harmonics we used was 337, which is very close tothe number 361 used by Li. Our calculation was per-formed on a 600-MHz Pentium III personal computer,which used 150 Mb of RAM and took 314 s for each fre-quency. It should be recalled that our approach treatsthe circular shape of the elements by use of exact analyti-cal expressions for the circular variation of e(r) in Eqs.(43) and (45), thus avoiding a time-consuming and com-plex staircase approximation.
We also show, in Fig. 2, the convergence of our ap-proach for that circular case. Note that convergence is asrapid as that achieved by Li14 in his Fig. 7. Further-more, his results (marked by crosses in Fig. 2) are veryclose to ours. We recognize that our choice of(gx , gy , ez) or (ex , ey , ez) expansions is not based onsolid analytical ground. However, it seems to providegood convergence and accuracy for most cases involvingcommon optical materials.
What is more important is that we have applied our ap-proach to situations for which no solutions have yet beenreported. These include structures with relatively manylayered regions, of which one or more have skewed lat-tices that contain nonrectangular elements. As a par-ticularly suitable illustration, we considered thequantum-well infrared photodetectors29,30 configurationshown in Fig. 3. This structure has a high measure ofcomplexity because (a) it contains five distinct layered re-gions bounded by semi-infinite substrate and cover re-
Table 1. Diffraction Efficiencies (in %) for Diffraction Orders Reflected by the Circular Post Gratinga
a The values in parentheses are those given by Li14 in his Table 2.
Lin et al. Vol. 19, No. 10 /October 2002 /J. Opt. Soc. Am. A 2013
gions, (b) the periodic quantum-well regions of thicknesstq are characterized by a uniaxial dielectric medium hav-ing ez Þ ex 5 ey , with ez being complex to account for thephotoelectric effect, (c) the periodic cylindrical posts canhave square, circular, or triangular cross sections, and (d)the lattices are generally skewed at an angle u as shownin Fig. 1.
In this context, we examined the dependence of the nor-malized scattered power pscat 5 Pscat /P inc on the latticestructure and on the shape of the posts. Here Pscat5 Ptrans 1 Prefl is the sum of the transmitted and the re-flected powers, and P inc is the incident power. The trans-mitted power is given by
Ptrans 5 R(K
@ f K,x~a ! f K,y
~a ! #* Y K~a !S f K,x
~a !
f K,y~a ! D , (70)
where R denotes the real part and a refers to the air re-gion. Similarly, the reflected power is given by
Fig. 2. Convergence of diffraction efficiency versus truncationorder N for several diffraction orders in the case of Example 2reported by Li.14 The results shown in his Table 3 are indicatedhere by crosses.
Fig. 3. Radiation incident on a 3D structure with a complex ge-ometry: (a) cross section in the yz plane, (b) three differentshapes in the xy plane for the posts considered here.
Prefl 5 R(K
@ gK,x~s ! gK,y
~s ! #* Y K~s !S gK,x
~s !
gK,y~s ! D , (71)
where the superscript (s) refers to the substrate region.The incident power is given by
P inc 5 @ f 0,x~s ! f 0,y
~s ! #* Y 0~s !S f 0,x
~s !
f 0,y~s ! D . (72)
For each of the three different types of posts shown in Fig.3(b), we have computed pscat as u varies from 0 to largervalues; i.e., the structure starts with a square lattice(with a2 perpendicular to a1 and a1 5 a2) after which they component of a2 remains fixed but a2 shifts along the xdirection so that a2 5 a1(x tan u 1 y). As u is increased,the lattice thus becomes progressively more slanted untilu 5 tan21(1/2). At this angle, every row of lattice pointsalong x is shifted exactly by a1/2 with respect to its adja-cent rows. Further increase in u does not lead to any newlattice structure; instead, the lattice gradually returns tothe original square form as u reaches p/4. Notice thatthe area of the unit cell is always given by a1
2, indepen-dent of u.
We have thus evaluated pscat versus u for unpolarizedwaves incident normally from the substrate. For thispurpose, the incident wave was assumed to be linearly po-larized, and for every lattice structure and post shape,pscat was calculated for polarization angles f and then av-eraged over 0 < f < p. The pertinent results are givenin Fig. 4. Specifically, Fig. 4(a) shows pscat versus u for anincident polarized (f 5 0) wave, Fig. 4(b) depicts pscatversus f for polarized waves and fixed u 5 13.5°, andFig. 4(c) yields the averaged (unpolarized) results as uvaries. It is important to note that the lowest value ofpscat is obtained for square lattices, irrespective of thepost shape. Of the three different shapes explored here,the circular post always gives smaller values of pscat thanthe square and triangular posts, and the triangular postalways yields the highest values of pscat . It is also inter-esting to note that, for performance purposes, the quan-tum efficiency QE of quantum-well infrared photodetec-tors is proportional to the normalized absorbed powerpabs 5 1 2 pscat . Hence for the shapes considered inFig. 3(b), the circular post would provide the highest QE,followed by the square and then the triangular posts. Inthis example, there is only one transmitted order andfrom five to seven reflected orders, depending on the angleu.
To assess the physical plausibility of these results, werecall that on the one hand, diffraction increases if thescattering objects have sharper edge boundaries, and onthe other hand, scatterers with higher degrees of symme-try generate less diffraction for incident unpolarizedwaves. Hence the highly symmetric square (u 5 0) lat-tices in Fig. 4(a) account for minimal scattering comparedwith any skewed (u Þ 0) but less symmetric lattice.Furthermore, among all shapes of individual scatterers,the smooth circular form scatters the least, followed hereby the square posts and then by the sharper triangularposts. We thus find that the results in Fig. 4 are fullyconsistent with the physical intuition afforded by consid-
2014 J. Opt. Soc. Am. A/Vol. 19, No. 10 /October 2002 Lin et al.
ering edge scattering in conjunction with the symmetriesof both the lattice and its internal components.
6. CONCLUSIONWe have presented a general rigorous solution to planewaves scattered by 3D periodic structures containing ar-bitrary combinations of homogeneous and periodic layersinvolving biaxial dielectric materials any one of which canbe transparent or absorbing. Our general method ap-plies to arbitrary lattice structures as well as to arbitrarydielectric variations within the unit cell. The approach isbased on taking advantage of vectorial descriptions forboth the real (space) lattice and the reciprocal (wave num-ber) lattice and on using modal fields inside each of thelayers. To obtain those fields within a periodic dielectriclayer, the relevant Maxwell equations were converted into
Fig. 4. Normalized scattered field pscat versus u for unpolarizedwaves at a wavelength of 8.3 mm, for the post shapes shownin Fig. 2(b). The cross-sectional area of the post is 4 mm2 forall three post shapes. The dimensions are a1 5 3.5, tc 5 0.1,tq 5 1.15, tb 5 2.0, ts 5 0.1, and d 5 2.6 mm. The upper re-gion is assumed to be air. The quantum-well region is modeledas an absorbing biaxial material, with dielectric constantsex 5 ey 5 10.43 and ez 5 10.43 1 i. The dielectric constant ofthe stop-etch layer is isotropic and has a value of 9.956. Thesubstrate and all the remaining layers are also isotropic, andtheir dielectric constant is 11.156.
algebraic ones by expanding the fields and the dielectricfunctions as Fourier series and then solving for the result-ing eigenvalue problem. The pertinent boundary condi-tions were then treated by using a general transmission-line formalism. Application of this approach providesaccurate results for the scattering and absorption proper-ties, which can be obtained with relatively modest compu-tational resources.
Numerical results obtained by our method have accu-rately reproduced data that had been reported for rela-tively simple configurations in previous papers. In addi-tion, we have shown that our approach can addresscomplex periodic structures that have not been reportedby other methods. Because it offers a basic analyticalrather than numerical procedure to treat situations thatinvolve multiple layers and gratings having a wide rangeof shapes, this approach is expected to provide a powerfultool for studying diffraction by complex 3D periodic struc-tures.
APPENDIX A: COMPARISON OF FIELDEXPRESSIONS FOR THE HOMOGENEOUSISOTROPIC AND UNIAXIAL CASESExpressions for the transverse electric and magneticfields in a homogeneous isotropic layer in Eqs. (25) and(26) appear to be rather different from the correspondingexpressions in Eqs. (35) and (37) for a homogeneousuniaxial layer. However, in the limit that e' 5 e i ,uniaxial layers become isotropic, and these expressionsmust be consistent. We discuss this limiting behavior indetail here.
Notice from Eqs. (31) and (33) that both eigenvalueskK,1 and kK,2 for a uniaxial layer depend on the dielectricconstants of the layer. In the limit that the layer be-comes isotropic, the eigenvalues become degenerate andreduce to those given by the isotropic case, i.e., kK in Eq.(21). In contrast, the eigenvectors for a uniaxial layer inEqs. (32) and (34) are independent of the dielectric con-stants of the layer, and they do not reduce to the eigen-vectors given in Eq. (24) for the isotropic case. Thereforethe expressions for the fields do not immediately corre-spond to each other. This happens because the eigenvec-tors for isotropic layers are usually chosen by using Eq.(24). However, we can use Eqs. (32) and (34) to choosethe eigenvectors to be the same as those for the uniaxiallayer. The electric field will then have the form
S Ex
EyD 5 (
Kexp~ikK • r!AK
3 H S fK,x
fK,yD exp~ikKz ! 1 S gK,x
gK,yD exp@ikK~t 2 z !#J .
(A1)
It is clear that in the isotropic limit the electric fields inthe uniaxial layer reduce to this alternate form for theelectric fields in the isotropic layer. Analogously, the ex-pressions for the magnetic fields will also correspond toeach other in the isotropic limit.
Expressions for the fields in a homogeneous isotropiclayer in Eqs. (25) and (26) in fact are equivalent to those
Lin et al. Vol. 19, No. 10 /October 2002 /J. Opt. Soc. Am. A 2015
in Eqs. (35) and (37) for a homogeneous uniaxial layer inthe isotropic limit, although they appear to have differentforms. We will now derive relations between the waveamplitudes in these expressions as well as the relationsbetween the two forms of admittance in Eqs. (27) and(38). In the following we use a prime to denote quanti-ties associated with an isotropic layer and a double primefor the corresponding quantities of a uniaxial layer.
In the isotropic limit, the expression for the electricfield in a uniaxial layer in Eq. (35) reduces to the form
exp~ikK! • S Ex
EyD 5 (
KAKF f K,x9 gK,x9
f K,y9 gK,y9G
3 S exp~ikKz !
exp@ikK~t 2 z !# D . (A2)
Comparing this expression for the electric field in an iso-tropic homogeneous layer as given in Eq. (25), we obtain arelation between the field amplitudes for the two cases asfollows:
F fK,x8 gK,x8
fK,y8 gK,y8G 5 AKF f K,x9 gK,x9
f K,y9 gK,y9G . (A3)
Recall that AK is an orthogonal matrix; thus its inversematrix is the same as its transpose matrix. The inverserelation is therefore given by
F f K,x9 gK,x9
f K,y9 gK,y9G 5 AK
TF fK,x8 gK,x8
fK,y8 gK,y8G . (A4)
In the isotropic limit, the expression in Eq. (37) for themagnetic field in a uniaxial layer takes the form
S Hy
2HxD 5 Ae0
m0(K
exp~ikK • r!AKYK9 F f K,x9 gK,x9
f K,y9 gK,y9G
3S exp~ikKz !
2exp@ikK~t 2 z !# D , (A5)
where YK9 is the admittance defined in Eq. (38). With thehelp of Eq. (A4), one can readily show that the magneticfields have exactly the same form as Eq. (26) for an iso-tropic layer and obtain the following relation between theadmittances for the isotropic layer and the uniaxial layer:
YK8 5 AKYK9 AKT . (A6)
We therefore recognize that the two representations ofthe wave amplitudes and the admittances are related toeach other through an orthogonal transformation.
APPENDIX B: INVERSE RELATIONBETWEEN eK,K8 AND gK,K8We show here that eK,K8 and gK,K8 are inverses of eachother only if all the reciprocal lattice vectors are included.This is true for any component of the dielectric constant,so we omit the index a below. Placing the expressions ofeK,K8 and gK,K8 into Eqs. (43) and (45), we have
(K9
eK,K9gK9,K8 5 Ecell
dr
VE
cell
dr8
V
e~r!
e~r8!
3 exp@2i~K • r 2 K8 • r8!#
3 (K9
exp@iK9 • ~r 2 r8!#, (B1)
where we have interchanged the order of the sum and thetwo integrals. If we sum over all the reciprocal latticevectors, we can use Eqs. (1), (3), (6), and (7) to show that
(K9
eK,K9gK9,K8 5 dK,K8 . (B2)
It is clear that we also have
(K9
gK,K9eK9,K8 5 dK,K8 , (B3)
so eK,K8 and gK,K8 are inverses of each other. Obviouslythis is true only if all the (infinitely many) reciprocal lat-tice vectors are included.
APPENDIX C: HOMOGENEOUS LIMIT OF APERIODIC LAYERWe are interested to see how the results derived in Sub-section (3.C) for a biaxial periodic layer reduce to thosederived in Subsection (3.B) for a homogeneous uniaxiallayer if we let ex 5 ey and take the limit that the layerbecomes homogeneous. In this limit, all the matrices ob-tained from the Fourier transforms of the dielectric func-tions become diagonal matrices, i.e., ex 5 ey 5 e iI andez 5 e'I. All the matrices in Subsection (3.C) can thenbe written as 2 3 2 block matrices, whose submatricesare all diagonal. This means that each K component ofthese diagonal matrices is uncoupled from any other.Hence the matrix equations can be written in terms of2 3 2 matrices, one for each component of K, as in Sub-section 3.B.
More specifically, we see that the mode index m can beseparated into two indices m1 and m2 . In the limit thatthe periodic layer becomes homogeneous, we have
km1→ kK,1 km2
→ kK,2 , (C1)
fm1→ fK,x , fm2
→ fK,y , gm1→ gK,x ,
gm2→ gK,y , (C2)
F ~PK,m1!x ~PK,m2
!x
~PK,m1!y ~PK,m2
!yG → ~AK! 5
1
kKF ~kK!x 2~kK!y
~kK!y ~kK!xG ,(C3)
and
Q → AKYK , (C4)
where YK is the admittance of a uniaxial layer given inEq. (38).
2016 J. Opt. Soc. Am. A/Vol. 19, No. 10 /October 2002 Lin et al.
APPENDIX D: DIVERGENCE-FREECONDITIONSAll the present results are consequences of Eqs. (11)–(13)obtained from Maxwell’s curl equations. The remainingtwo Maxwell equations that involve the divergence arenot explicitly being used. One may then questionwhether this second set of two equations holds at all. Itis known that for certain computational techniques inelectromagnetics, violation of these equations can causenumerical problems, such as spurious solutions and insta-bilities. We show here that the field solutions as devel-oped by using modal fields automatically obey the diver-gence equations.
We first check to see if our expressions for the fields in-deed obey the “ • D 5 0 condition. Using Eqs. (40) and(14), we have
“ • D 5 ]x(K9
eK9,x exp~iK9 • r!(K8
exp~ik8 • r!
3 (m
~PK8,m!xvm~z !]y(K9
eK9,y exp~iK9 • r!
3 (K8
exp~ikK8 • r!(m
~PK8,m!yvm~z !
2i(K
exp~ikK • r!(m
@~kK!x~QK,m!x
3 ~kK!y~QK,m!y#kmvm~z !. (D1)
In the two sums over K8 on the right-hand side, we letK9 5 K 2 K8, and instead of summing over K9 we sumover K. Then we carry out the partial derivatives to get
“ • D 5 i(K
exp~ikK • r!(K8
(m
H ~kK!xeK2K8,x~PK8,m!x
1 ~kK!yeK2K8,y~PK8,m!y 2 dK,K8@~kK8!x~QK8,m!x
1 ~kK8!y~QK8,m!y#km
k0J vm~z !. (D2)
In matrix notation, we note that
xexPx 1 yeyPy 2 ~ xQx 1 yQy!k
5 @ xex yey#P 2 @ x y#Q k, (D3)
where k is a diagonal matrix containing the eigenvalueskm . Using Eqs. (56) and (54) we can show that the aboveequation is zero, so that Eq. (D2) and therefore “ • D arealso zero.
Next we show that “ • H 5 0. Using Eqs. (41) and(14), we have
“ • H 5 iAe0
m0(K
exp~ikK • r!(m
$2~kK!x~QK,m!y
1 ~kK!y~QK,m!x
1 @~kK!x~PK,m!y 2 ~kK!y~PK,m!x#km%im~z !.
(D4)
In matrix notation, we note that
@ y 2x#Q 2 @ y 2x#P k 5 0, (D5)
which can be shown by using Eqs. (55) and (53). HenceEq. (D4) and therefore “ • H are zero. We thus concludethat the fields used here in our method automaticallyobey the two divergence-free conditions in Maxwell’sequations.
ACKNOWLEDGMENTSThe authors thank K. K. Choi of the U.S. Army ResearchLaboratory, Adelphi, Maryland, for helpful discussionsand guidance on the quantum-well photodetectors dis-cussed here. The present work was supported in part bythe U.S. Army Research Office under grants DAAH04-96-1-0389 and DAAD19-99-1-0126.
Corresponding author K. M. Leung can be reached byemail at [email protected].
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