Department of Electrical Engineering, College of Engineering, University of Osaka Prefecture, Sakai, Osaka,Japan
Received September 23, 1982
A method for analyzing slanted anisotropic gratings is presented. The propagation of electromagnetic waves inthe periodic anisotropic media is described by coupled-wave equations in a unified matrix form expressing the cou-pling of the space harmonics in the grating region. The solution of the equations is reduced to an eigenvalue prob-lem of this coupling matrix. Through introduction of the concepts of transmission and boundary matrices, the dif-fraction properties of general slanted gratings are obtained rigorously by systematic matrix calculations that areeasily implemented on a computer. The calculated results indicate that not only TE-TE or TM-TM but alsoTE-TM diffractions take place in general slanted gratings.
INTRODUCTION
The scattering and waveguiding properties of electromagneticwaves by periodic dielectric structures have been extensivelystudied in recent years. These periodic structures play animportant role in applications such as couplers, filters, andmany other Bragg-type devices in the optical and millime-ter-wave regions. Several authors have proposed differentmethods to analyze these periodic structures, which aresummarized in the coupled-wave and modal approaches.1-' 2
However, to the authors' knowledge, rigorous analysis hasbeen limited in the case of unslanted isotropic gratings.
Recently, Moharam and Gaylord13 applied the rigorouscoupled-wave approach to analyze the diffraction of an elec-trpmagnetic plane wave incident obliquely at a planar gratingbounded by two different media. However, their analysis islimited to the case of a sinusoidal grating oriented in the planeof incidence for a TE wave. Wagatsuma et al. 14 discussed theTE-TM wave coupling in a periodic optical waveguide whenthe waves propagated obliquely with respect to grating vec-tors. Their calculations are based on an approximate two-wave analysis. On the other hand, Yeh15 analyzed wavepropagation in birefringent layered media and showed theband structure peculiar to a S6lc layered medium.
The rigorous approach presented here analyies the dif-fraction properties of general anisotropic periodic mediawhose grating vectors have arbitrary orientation in three di-mensions. The method is general and straightforward andcan easily be extended to layered gratings. Therefore mostof the previous problems of gratings of this kind may betreated as special cases of the present analysis.
ELECTROMAGNETIC FIELDS IN THEGRATING AND UNIFORM REGIONS
We first conisider the diffraction of an electromagnetic planewave incident obliquely at a lossless anisotropic gratingbounded by two different isotropic media, as shown in Fig. 1.The Cartdsian-coordinate system is chosen such that the xaxis is normal to the interfaces, and all the field componentsare assunmed to have time dependence of exp(jwt).
The input and output regions (regions 1 and 3) are homo-
geneous dielectrics with relative permittivities E1 and e3, re-spectively. Without loss of generality, the plane of incidenceis confined to the xz plane with the angle of incidence 01.The incident wave has a general polarization with all the fieldcomponents having a space dependence of expVEjVko(xcos Oi - z sin Oi)], where ko = c o = 2ir/X and X is thefree-space wavelength.
The grating region (region 2) consists of an anisotropic di-electric with the grating vector k having any orientation inthree dimensions:
K = ixKx + iyKy + iK 2, (1)
with
IKI =K= 27r/A, (2a)
Kx = 2,r/Ax, KY = 2W/AY, K, = 2r/A,, (2b)
where ii (i = x, y, z) is the unit vector along the i axis, A is thegrating period, and Ai is that along the i axis, respectively.Throughout the paper we use the subscripts (i, j) as thecoordinate indices (1, m, n) as the space-harmonic indices, andthe (I, J) as the region indices. For simplicity, region indicesare usually suppressed unless otherwise stated. The relativetensor permittivity e of the medium is then given by
where Eip are the lth order Fourier coefficients known for agiven grating. For simplicity, the relative permeability of themedium is assumed to be 1, but it can easily be extended toA = (Aij), as given in Appendix A.
We now normalize the space coordinate by ko and put r =kor, x = kox, y = koy, and z = koz, respectively, where theoverbar indicates normalization by ko. Then Maxwell'sequations are expressed as
cur1V'"TE = -jV/'H, curlv/'H = jE(Y)VY0E,
Cy\
ft = hez
khyv
= (;1xj, (13)
where ei = ei (x) and hi = hi (x) are column matrices with el-ements eim (3) and him (Y), respectively. Then, substitutingEqs. (9a) and (9b) into Eq. (5), we get the infinite set of cou-pled-wave equations, which are written in the following matrixform, as derived in Appendix A:
(5) where
dYd-ft = jCf,
fn= Dft,
(14)
(15)
/ qc- I ey + P
s|ex (eXyY +S2
\ e - ezIEx eXy + qs
qC- 1e ~elq + p
setI qXX_C C iq
qeC lX- 1,z -qe-ls
tyex -,Exz - eyz - sq -eyx -S
SeXXEXZ + P -se-1s + 1
CZZ - ezx lexz - q 2 eZXex S + p
where Yo = liZO = V`;7/I and D (-exy -Clq -e-1exz e1s)-s S q 0 ~
Eij(Y) = Z Eiji exp(JlnK * r),
nK = ixP + iyq + izs;
(6)
(7)
I nK| = nK=X/A,
p =X/A, q = X/Ay, s = X/A,
(8a)
(8b)
and nK is considered to be an effective refractive index Qf thegrating. From the Floquet theorem, electric and magneticfields can be expressed as
VTY0E = Z em (x)exp (-jnm- r),m
V'-H = _ hm(Y)exp(-jnm * r),m
em(Y) = ixexm(Y) + iyeym(Y) + Izezm(.),
hm(Y) = ixhxm(Y) + iyhym(Y) + izhzm( (),
(9a)
(9b)
(17)
Here, eij = (eij,ni) are (2m + 1) X (2m + 1) submatrices whoseelements are given by
eijnl = Eij,n-1h (18)
and the other (2m + 1) X (2m + 1) submatrices are diagonal,that is,
P = (On1PI), q = Onjqj)
s = (6 nlsl), 1 = (Onl), (19)
where 6 nl is the Kronecker delta. The coupling matrices Cand D characterize the coupling between all the space har-monics of the TE and TM waves in the general anisotropic-grating region. For the usual isotropic gratings, cij = bije, andtherefore C is reduced to
P qclq- 1
C (2e p
qs 0
0 -qes\-sq 0
p 1- Se-1 sE-q2 p
(20)
Moreover, when the grating vector is in the xz plane, q = 0,C is then further simplified to
(lOb)C = (CTE CO
0 O TM1and
with
+ +
so = V,/ sin Oi,
Pm = mp, qm = mq, Sm =SO+ ms.
(11)
(12a)
(12b)
The index m under the summation sign is understood to runover all values of m = 0, 11, 2....
We now express the tangential and normal components ofthe fields at the interfaces as
(21)
with
CTE = (S2- )1' CTM = (P - SC-I) (22)
In this case, Eq. (13) is separated in two independent equa-tions:
d-fiTE = XCTEfATE,
- ftTM = jXTAfTM,dx
fiTE = ~jf hM
fiTM = ()
(23a)
(23b)
with
here
(16)
where
K. Rokushima and J. Yamakito
Vol. 73, No. 7/July 1983/J. Opt. Soc. Am. 903
Therefore, TE and TM waves cannot be coupled by thegrating, which is the case treated previously only for TEwaves.' 3
In the uniform isotropic regions (regions 1 and 3), e in Eq.(20) is a diagonal matrix, that is, e = El, and p can be set equal
to zero. Therefore, C becomes
0 q 2 /E - 1 0 -qs/E
cu = s2 - El 0 -sq 0 (24)
0 sq/E 0 1 -S2
/E
qs 0 cl - q 2 0 /where all the submatrices are diagonal. Then each space-harmonic field separately satisfies Maxwell's equations so thatEqs. (14) and (15) are separated into (2m + 1) independentequations:
d- tm = jCXft, fm = Du fim,nm m tm (25)
/m Om
Vmil = :Smtm Ym2 ) = Eqm/#m I
Vm3 J -qm Pm4 Sm
\qm m \FESm/lm
Tu = (Vml, Vm2, Vm3, Vm4),
(32)
(33)
where
=(-2 )1/2,qm = (e - ntm
qm= qm/nim,
nt2m = (q + S2 )1/2 (34a)
Sm = Sm/lnm. (34b)
It can be seen that the characteristic field correspondingto the eigensolution Vmi exp(-jtmY) constitutes the TE wave(with respect to the plane of the mth-order diffraction wave)propagating along the direction of nu = 'xAm + 'yqm + lzSm,
since, from Eqs. (25) and (26), the characteristic field com-ponents are given by
(26) where field amplitudes are normalized to I eC~mi = IemlI = 1,
hmil =,and em, -nu = hm, -nu = em, -hm =0 that is,em,, hmi, and nu are mutually perpendicular.
Similarly, the characteristic field corresponding to Pm2
(27a) exp(HjWmO) constitutes the TM wave (with respect to theplane of diffraction) whose field components are given by
Cm2 = ixexm2 + etm2 = -xnjm/(m + (7y4m + Iz m),
(27b)
hm2 = him2 = (-l1ym + 7 4m)El/m.
(36a)
(36b)
MATRIX FORMULATION OF THE SOLUTION
Characteristic Fields in the Grating and Uniform RegionsThe solution of coupled-wave equation (14) is reduced to aneigenvalue problem of the n X n matrix C, where n = 4(2m +1). Let Kn be an eigenvalue determined from the character-istic equation
det(C - K1) = 0; (28)
the corresponding eigenvector vn (with elements Pnl ) and thediagonalizer T = (v1, v 2 ... vn) for C can then be obtained. Wethus transform fi to g by the following equation:
ti = Tg,
so that Eq. (14) is transformed into
dd- g = JKg,
(29)
(30)
em2, hm2, and nu are also mutually perpendicular, and theiramplitudes are normalized to I eim2l = 1, Iem2l = \/E/I m 1,
and I hm2I = Eli Im |. The remaining characteristic fields alsoconstitute similar TE and TM waves that propagate in thereverse direction along the x axis. For m = 0, qm = 0 and gm= 1. The characteristic fields are then TE and TM waves withrespect to the xz plane (the plane of incidence).
From these results, it can be seen that the diagonalizercorresponding to Cu is given by
S
Tu = (-q
dt
q q
Ef14~ -St -eqtl I1'
§ -q s J-E5t-l -qt ES~ /
(37)
with diagonal submatrices q = (Oniqi), s = (Onii), and t =(
3nl l). Here, the corresponding g is expressed as
where K = (n3jKj) is a diagonal matrix and g is a column matrixwith elements gn. Since Eq. (30) has a solution of the formgn()= exp(jKnx), f^ has an eigensolution vn exp(jKnx-). Inthe grating region, vn contains elements vni so that the char-acteristic field corresponding to the eigensolution containsan infinite set of space harmonics. This is in contrast to thefields in the uniform regions.
In the uniform regions, eigenvalues, eigenvectors, and di-agonalizers for Cu are explicitly given by
KmI = Km2 = im, Km3 = Km4 = (m 3
(38)
Eg+\
g gl = 'Mgg- ),
g-
withEg'= = (Egg... Ego Eg+)t
mg+ = ft+.. Mgo.. mg ±)
(39a)
(39b)
where the pluses and minuses of the superscripts refer to thewaves traveling along the +x or the -x directions and E andM refer to TE or TM waves, respectively.
0
qs -m
m sm
K. Rokushima and J. Yamakito
(31)
904 J. Opt. Soc. Am./Vol. 73, No. 7/July 1983
Transmission and Boundary MatricesWithin region I (I = 1, 2,3), the solution of Eq. (30) is givenin matrix form as
g (y) = Ui(y - Yo)g1 o), (40)
with
UJ( - Yo) = expUiKI(Y - Yo)], (41)
where expUKI(Y - Yo)I = 1an1 expUjKnI( - xo)]) is a diagonal
matrix and g1 (7o) is a constant column matrix at x = Yo.
UI(Y - Yo) is the transmission matrix representing thetransverse propagation of g1 along the direction of the x axisin region I.
At a boundary surface (x = 0 or x = d) separating thegrating region from the uniform region, tangential componentsof the fields are continuous across this boundary. Therefore,from Eqs. (9a), (9b), and (13), EXP(-jpiY)fij should becontinuous at the boundary interface, where
Here
93 = 0.
Therefore diffracted waves are given by
= W-7g, g = W 2W4 gj.
(50)
(51)
Since the mth-order TE and TM diffraction powers alongthe +x or -x direction in region I (I = 1, 3) are
Ep I = Re Qlm,) I 19' I m
Mp 1 = Re(EI/tim) Mg'I ,
(52a)
(52b)
with 4Im = (el - - S2 )1/2, the mth-order diffraction ef-ficiencies of the reflected and transmitted waves i7' and ?7tare given by, for the incidence of a TE wave,
EEr = Eptm/i10O
EE I = Ep m/410,
EMvr = Mptm/i10O
EM 1 = Mp3-m/610O (53)
exp(-jply) 0
exp(-jpjy)
0 exp (-j pl)/
and for the incidence of a TM wave,
ME-qr = ^loEpl/l,(42)
is a diagonal matrix with diagonal submatrices exp(-jpiY)= [6ni exp(-jpnjY)]. Boundary conditions at x = O and x =d are then given, from Eq. (29), as
Tjugl(0) = T2"g 2(0), (43a)
EXP(-jp 2 d)T2 g2 (d) = T3Ug3 (d). (43b)
These boundary conditions are simply expressed by theintroduction of the boundary matrices Bij relating g1 to gjat the interface:
where, for instance, the superscript EM signifies TE - TMdiffraction. For lossless gratings, the power-conservationrelation requires that 2m;EM(flX + nq) = 1, which may beused as a check for the numerical calculations.
It can be seen that not only the usual TE-TE or TM-TMdiffractions but also TE-TM diffractions take place withgeneral gratings.
For the incidence of a general polarized wave,
g1 = (O .. a . .. 0) t, Mg- = (O. .. b ... out, (55)where I a 12 + I b 12
= I and a/b is determined from a given po-larization. Diffracted waves are also given by Eq. (51), anddiffraction efficiencies can be given in a similar way. Forinstance, the mth-order diffraction efficiences of the reflectedwaves E-qr and M1,, are given by
By using these transmission and boundary matrices, thematrix equation relating g1 to g3 at both interfaces of thegrating region is given by
E r = Ep Im/p - M r = MPm l/p-
where
-=l1 (W3 WI)(g3) -W
with
W = B 12 U 2 (d)B 2 3 ,
(46)
(47)
where W is the transfer matrix of the grating and Wk (k = 1-4)are its submatrices.
WAVES DIFFRACTED BY GRATINGS
If a unit TE or TM wave is incident in the xz plane from region1, diffracted TE and TM waves will be given by Eq. (46),where, for the incidence of a TE wave
Mg- = 0, (48)
and for the incidence of a TM wave
Mg- = (O .. . .1... O) t Eg- = 0. (49)
(56)
(57)
EXTENSION TO MULTILAYERED PERIODICSTRUCTURES
For simplicity, we have considered single-layered gratings sofar. However, the results are easily generalized to a multi-layered grating having the same period in the tangential di-rection, as shown in Fig. 2. This extension may be useful, forinstance, for application to arbitrarily shaped gratings bypartitioning them into fine layers and approximating each ofthese by rectangular profiles.' 6 By using the transmission andboundary matrices, the matrix equation relating g, to gN inthe layered grating can be similarly given by
91 = WgN- (58)
W is the transfer matrix of the layered grating, which is ex-pressed by
W = B12 U2(d 2)B 23 .. UN-1(dN-1)BN-1,N, (59)
EXP(-jpiY) =
K. Rokushima and J. Yamakito
p- = I a 12t1o + I b 1 2(1/�10.10
Vol. 73, No. 7/July 1983/J. Opt. Soc. Am. 905
X- -1 +
+1 +1
d 2 . Ki Zd3 Kd N-1$ \ tN- 1 N- I
9+ 4 -N -1 -1 N
Fig. 2. Geometry of multilayered anisotropic grating.
where B-11, is the boundary matrix at the interface of theadjacent layers and is given by
Bi-i, = T-11 EXPU(p1 1 - pj)d1 1]TI. (60)
Therefore calculations can be performed systematically as insingle-layered gratings without increasing the matrix di-mension. This method of solution is the extension of ourprevious analysis of anisotropic layered waveguides for layeredgratings.17,18
NUMERICAL RESULTS
For comparison with the results in most of the previous work,we consider the problem of diffraction by sinusoidal gratingsbounded by two different media. The relative permittivityin the modulated region eij (j) in Eq. (6) is then given by
eij(r) = eij + Acij cos(nK. *)
= eijIl + 5[exp(jnK- Y) + exp(-jnk * F)]},
0.01 are not shown in the figure. In order to avoid overflowand round-off error problems, quadruple and double precisionhave been used in all the computations. The error in thepower-conservation relation was of the order of 10-9 in all ourresults.
Table 1 shows, as an example, for the incidence of TMwaves in Fig. 5(a), the accuracy of the solution by truncationof the matrix C, where mi is the order of space harmonics takeninto calculations, that is, m = Ot, .... ., L. From this tableit can be seen that the solution has almost converged graphi-cally at mT = 3. Since the other examples showed similar re-sults, later calculations have been performed by taking cal-culations up to m = 3. Table 2 shows similar results but withthe modulation index 6 as a parameter. From this table theorder of harmonics that should be taken into calculation fordesired accuracies may be estimated.
For the incidence of a TE wave in Fig. 5(a), the result ofcourse agrees graphically with that of Moharam and Gay-lord.1 3 For the incidence of a TM wave, the result is slightlydifferent from that of a TE wave. Figure 5(b) shows an ex-ample of a grating whose vector is not in the plane of incidence(q #- 0). It can be seen that the incident TE wave is diffractednot only in the TE waves but also in the TM waves. Thisshows that the gratings with q 5^ 0 have TE-TM-conversioneffects, which may have some use as mode-conversion de-vices.
We next consider the problem of anisotropic gratings withe2 having polar-type anisotropy, that is,
EXX
e2 = 0
0
0 0
EyZ Ezz
(65)
For simplicity we assume that the grating vector is in the xz
(61)
where 6 = Aeij/2Eij and eij reduces to bije for isotropic gratings.The waves are incident in the xz plane with the angle of inci-dence Oi as shown in Fig. 3; the grating vector with angles 0 and0 is shown in Fig. 4, where
p = nK cos 0, q = nK sin 0 cosqX, s = nK sin O sin 0.(62)
We first treat the problem of isotropic gratings with scalarE2 . e in Eqs. (20) and (22) is given by e = E2[E'41], where
I6 0 01 6. 0\
(e1; 6 0 i ) 6 AE/2E2 . (63)
0 6 1 /The Mth-order Bragg condition is satisfied by
MnK = -2-V(cos 0r cos 0 + sin Or sin 0 sin /), (64)
where 0r is the refraction angle determined by A/; sin 0r =
\I+ sin Oi.Figure 5 shows the diffraction efficiencies of the transmitted
waves for the incidence at the first-order Bragg condition,where El = E2 = E3, 26 = ICA2 = 0.121, with (a) 0 = 900 (q = 0)and (b) 0 = 450 (q 5 0). The diffraction efficiencies less than
region 1 E l
region
region 3 E 3 +2
-1 dtionFig. 3. Geometry of planar-grating diffraction.
p
Fig. 4. Direction of grating vector.
z
K. Rokushima and J. Yamakito
906 J. Opt. Soc. Am./Vol. 73, No. 7/July 1983
1.0
0.8
0.6
0.4
0.2
0.0
- TE-TE
TE-TM
m = 00z
r-4
z~_00
0.0 1.0 2.0 3.0 4.0 5.0d /A
(a) q =0
0.0 1.0 2.0 3.0 4.0 5.0d/A
(b) q * 0Fig. 5. Diffraction efficiencies of the isotropic grating for the incidence of TE and TM waves. Here Oi = 1600,0 = 60, M =1, and 26 =0.121.The grating vectors are (a) in the plane of incidence (I = 900, q = 0) and (b) not in the plane of incidence (0 = 450, q #d 0).
Table 1. Accuracy of the Diffraction Efficiencies of the Slanted Grating for the Incidence of TM Wavesam 2N+ 1 MM t4 MM7t3 MM-t 2 mmt 1 MM MM1t MMmt2
plane (q = 0). Coupling matrix C in Eq. (16) is then reducedto
-1 0 0
p - (YZ 0
o p - -sfq s ,o ez - q2 rp
(66)
with eij = efj(en') and 6 = Eiei/2eij.We further assume thatTE and TM waves are phase matched, as shown in Fig. 6.The Mth-order Bragg condition for TM waves is satisfiedby
MnK = -2VU Exx cos 0 cos 0 r + ezz sin 0 sin 0 r (67)withXX C052 6 + Ezz sin2 s
with V 7e, sin 0r = \/-eI sin 0,.
0z
14~
F4.
z0'-4
0
1-4
- TE-TE1.0
0.8
0.6
0.4
0.2 [-+I - - -00. -1 -' - --. - =F-1
- ey0
K. Rokushima and J. Yamakito
Vol. 73, No. 7/July 1983/J. Opt. Soc. Am. 907
Fig. 6. The first-order Bragg condition for TM waves in a polar-typeanisotropic grating. TE and TM waves are phase matched in themedium.
Figure 7 shows the diffraction efficiencies of transmittedwaves for the incidence of TE waves with 6i = 1600, el = E2 =
E3, M = 1, and
1.121
f2 =` E2 0
0
0 0
1 0.121 ,
0.121 0.986
where (a) 25 = 0-and (b) 23 = 0.121. Diffraction efficienciesless than 0.01 are not shown in the figure. When region 2 iswithout modulation (6 = 0), TE and TM waves exchangepower by the presence of the perturbation Eyz and, under aphase-matched condition, a complete spatially periodictransfer between TE and TM waves takes place, as in Fig. 7(a),which is a well-known result. When region 2 is furthermodulated, an incident TE wave is converted to diffracted TEand TM waves, as in Fig. 7(b). It can be seen that an incidentTE wave is equally converted to zeroth- and first-order TMwaves at d/A - 1.67 and to first-order TE and TM waves atd/A 2.69, with all other diffracted waves almost suppressed.
These may be diffraction properties peculiar to anisotropicgratings.
CONCLUSIONS
The diffraction properties of general slanted gratings havebeen analyzed by using a general coupling matrix that ex-presses the behavior of the space harmonics propagating inperiodic anisotropic media. The medium may be anisotropicor isotropic. The direction of the grating vector, the modu-lation of the medium, and the polarization of the incidentwave are all arbitrary and are specified by the dimensionlessparameters nK, eij, and so. The solution is reduced to an ei-genvalue problem of this coupling matrix whose elements aregiven by a simple and unified form. By introducing theconcepts at the transmission and boundary matrices, the re-sults are generalized to multilayered gratings.
APPENDIX A. DERIVATION OF THECOUPLING MATRICES
By substituting Eqs. (9a) and (9b) into Eqs. (5), multiplying(68) exp(jnm * r) on both sides, and integrating over the yz plane,
we get the infinite set of coupled-wave equations
Oeym -.- hmay -iPmeym + jqmexm =-ihzm,
aezmaezx - JPmezm + jSmexm = jhym,
ahYm - ipmhym + jqmh.xmdhym
-JPmhzm + iSmhxmay
= i 2 Y_ ezi,l-meil,I i
= -i E E Eyj,-mei1,I i
qmezm - Smeym = hxm,
-qmhzm + smhym = EZ E exi,l-meil.I I
(Al)
(A2)
(A3)
(A4)
(A5)
(A6)
TE-TE
--TE-TM
F-4
r-IC4 .
0F-4
C:.
1.0
0.8
0.6
0.4
0.2
I " -E I o.o LU
1.0 2.0 3.0 4.0 5.0 0.0d/A
(a) 6 = 0
Fig. 7. The diffraction efficiencies of the anisotropic grating for the incidence of TE waves.grating regions are (a) unmodulated (6 = 0) and (b) modulated (28 = 0.121).
T TE-TE
------ TE-TM
1.0 2.0 3.0 4.0 5.0d/A
(b) 6 # 0Here Oi = 1600, 0 = 600, (/ = 900, and M = 1. The
1.0
0.8
0.6
0.4
0.2
UzUzH
0E_U
I-I
CD
44. I
0.0 i-L
0.0
K. Rokushima and J. Yamakito
908 J. Opt. Soc. Am./Vol. 73, No. 7/July 1983
Equations (A5) and (A6) can be solved for exm and hxm, andthen, substituting these into Eqs. (A1)-(A4), we get the cou-pled-wave equations (14) and (15) in matrix form with C andD given by Eqs. (16) and (17).
For magnetic media such as ferrite with relative tensorpermeability A = (4jj) and relative scalar permittivity E, Eqs.(5) become
The coupling matrices in this case are similarly given by
p + AXL ls qc-lq + sLz~kxAlxz -Lz - AzXAx1 q
sAu - 1 s Cp + sAux I Yxz -sg xxqI Ayx_ I -1q + yLz - AyXA Lxz p + AyxAxlq
\q~AxiS q~Axxgxz e - qAlq
(_Ax is -A lAxz yx q - -1 -1xy
q/.Lxx uX
-qecls + lazxijxh.Lx
s~s;Axx I$y-xyxy xISx1yy - S(-'S - AYqXjX y
P + q /1XXAxlyy
APPENDIX B. WAVES GUIDED BY GRATINGS
Dielectric gratings can also support surface waves and leakywaves as periodic waveguides. Because of the energy loss thatis due to scattering by the grating region, the propagationfactor sj generally has a complex value, that is,
so = so'-jso".
Moreover, guided-wave conditions require that
g1 = gN = 0.
Therefore, we obtain from Eq. (58)
(gl) (W1 W2)({ °
\ 0 J = Z3 W4~1_N vg
(B1)
(B2)
(B3)
This is the transverse resonance condition for the periodicwaveguides, from which characteristic equation can be ob-tained as follows:
det(W4 ) = 0. (B4)
The modal analysis of the periodic waveguides is reducedto determining the propagation factor so = so' - so" satisfyingEq. (B4). If so is determined, gj and g- can be obtained fromEq. (B3), and all the field components in the grating regionare determined by use of Eqs. (9a), (9b), and (29).
ACKNOWLEDGMENT
The authors wish to express their thanks to H. Teraguchi andT. Takagi for their assistance in the numerical calculations.Thanks are also due S. Mori and M. Kominami for theirvaluable suggestions.
REFERENCES
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