1. INTRODUCTIONPlanar components that incorporate gratings to transferenergy from one dielectric waveguide to another havebeen developed for use in directional couplers, filters, la-sers, and other integrated-optics devices.1,2 The wavecoupling process in those applications has been analyzedmostly in terms of coupled modes, which involve varyingamounts of approximation.3–5 However, more accuratetechniques are often required for the proper evaluation ofsignificant features, such as the partition of energy be-tween the coupled waveguides, the power losses that aredue to radiation, and other operational characteristics.5–8
Highly accurate computational methods based on exactsolutions have therefore been employed,9,10 but their nu-merical nature cannot easily be translated into practicaldesign criteria.We present here an alternative approach by using rig-
orous modes that can accurately assess the characteris-tics and the performance of grating-assisted waveguidecouplers. These modes are exact solutions of the perti-nent boundary-value problem posed by a wide range ofplanar periodic configurations,11,12 of which grating-assisted couplers form a special class. Using this ap-proach, we have already reported13 that optimum powertransfer occurs under conditions that may be differentfrom those predicated by coupled-mode theory. In thispaper we discuss the advantages of the modal approachby presenting characteristic performance curves for typi-cal grating-assisted couplers. In addition, we determinethe power-transfer mechanism and compare it with thatgiven by coupled-mode methods. Our results reveal thatoptimum coupling conditions, maximum power-transfercapabilities, and other operational aspects of grating-assisted couplers are strongly affected by the field distri-bution in the cross section of the guiding configuration.
2. FIELD SOLUTION IN THE GRATINGREGIONGrating-assisted waveguide couplers consist of stackeddielectric layers of height tj , where j 5 1, 2, . . ., J iden-
0740-3232/96/1202403-11$10.00
tifies specific layers, which are located between semi-infinite substrate ( j 5 0) and cover ( j 5 J 1 1) regions.In one of the layers, which is identified by some j 5 p, therefractive index varies periodically with z, thus forming agrating of period L. Such a typical configuration havingJ 5 4 and p 5 3 is illustrated in Fig. 1(a), where thethird layer is a grating whose dielectric constant alter-nates between the values e2 and e4 of the adjacent regions.The grating may be located at another ( j Þ 3) level, sothat the index p can generally take on any value1, 2, . . ., J. For greater generality, the dielectric con-stants in the grating are allowed here to alternate be-tween any two values ep1 and ep2, as shown in Fig. 1(b).The permittivity of all media in Fig. 1 may generally be
complex such that, for a time dependence exp(2ivt), wehave e j 5 e j8 1 ie j9 , with e j8 . 0. In the bounded layerse j9 is then positive, zero, or negative if the jth medium islossy, lossless, or gainy, respectively. However, the openregions must be passive to ensure finite fields as x→6`,so that e j9 > 0 for both j 5 0 and J 1 1. The layers hav-ing larger dielectric constants, e.g., j 5 1 and 4 in Fig.1(a), act as input and output waveguides. Those guidinglayers usually terminate at z 5 0 or L as in Fig. 1(a), sothat they overlap only in the periodic domain0 < z < L, which is referred to as the coupling region incoupled-mode theory. It is important to observe that,while other methods impose serious limitations on thegrating parameters (e.g., see Subsection 4.E on p. 976 ofHuang’s comprehensive review paper5), the modal ap-proach used here is not restricted to either small gratingheights or large separation distances, i.e., tp and t2 in Fig.1(a) can be arbitrary.For two-dimensional (]/]y 5 0) situations any field
component inside the periodic region 0 < z < L can bedescribed in the rigorous Floquet-type form
F~x, z ! 5 exp~ikz ! (n
Wn~k; x !exp~2inpz/L!, (1)
where the time dependence was suppressed and the sum-mation ranges over all n 5 0, 61, 62, . . .. Here k is the(generally complex) propagation factor, which must be de-
2404 J. Opt. Soc. Am. A/Vol. 13, No. 12 /December 1996 S. Zhang and T. Tamir
termined for every mode of interest. For gratings havingsymmetric rectangular profiles as in Fig. 1(b), the func-tions Wn(k; x) can be found exactly.12,14
We conveniently subdivide the vertical axis into localsegments xj extending from the bottom to the top of eachbounded layer, so that 0 < xj < tj for all j Þ 0 orJ 1 1; in the open regions we take x0 5 x with t0 5 0 inthe substrate and xJ11 5 x 2 ttot in the cover, where ttotis the sum of all tj . The fields in the homogeneous ( jÞ p) regions can then be written as
Ej 5 (n
$fjn exp~ikjnxj! 1 gjn
3 exp@ikjn~tj 2 xj!#%exp~ikznz !, (2a)
Hj 5 (n
yjn$fjn exp~ikjnxj! 2 gjn
3 exp@ikjn~tj 2 xj!#%exp~ikznz !, (2b)
where f0n 5 gJ11,n 5 0 in the open regions. Here Ej andHj refer, respectively, to tangential electric Eyj and mag-netic Hzj components for TE modes, or tangential electricEzj and magnetic 2Hyj components for TM modes, and
kzn 5 k 1 2np/L, with k 5 b 1 ia, (3)
kjn 5 kjn8 1 ikjn9 5 Ako2e j 2 kzn2 for all j Þ p,
(4)
yjn 5 H kjn /vmo ~TE!
veoe j /kjn ~TM!, (5)
Fig. 1. Periodic configurations: (a) typical grating-assistedcoupler, (b) general geometry of a rectangular grating layer form-ing the periodic part of the coupler. Unless otherwise stated,the numerical examples given in this paper assume that e05 e2 5 ep2 5 3.02, e1 5 3.52, e4 5 ep1 5 3.22, e5 5 1.0, t15 0.22 mm, t2 5 0.55 mm, t3 5 tp 5 0.05 mm, t4 5 0.45 mm, andL1 5 L2 5 L/2 5 5.374 mm.
where ko 5 2p/l 5 vAmoeo, with l, mo , and eo denotingwavelength, permeability, and permittivity in vacuum,respectively. To satisfy radiation conditions and to mini-mize computational difficulties, we define the squareroots in Eq. (4) such that
kjn8 . 0 if ubnu < Re~koAe j!, or kjn9 . 0 otherwise,(6)
where Re(u) refers to the real part of u.The fields in the grating ( j 5 p) layer are given12 by
Ep 5 (m
$fpm exp~ikpmxp! 1 gpm
3 exp@ikpm~tp 2 xp!#% (n
anm exp~ikznz !, (7a)
Hp 5 (m
ypm$fpm exp~ikpmxp! 2 gpm
3 exp@ikpm~tp 2 xp!#% (n
bnm exp~ikznz !, (7b)
with
ypm 5 Hkpm /vmo
veo /kpmgp
~TE!
~TM!, (8)
where gp denotes the average of 1/e(z). The quantitieskpm , anm , and bnm have known characteristic values thatsatisfy the wave equation inside the periodic region.These values can be determined analytically12,14,15 for allgratings whose periodic variation e(z) 5 e(z 1 L) is in-dependent of x. In particular, kpm is given not by Eq. (4)but by a more complex dispersion relation in the form ofHill’s determinant.14 That dispersion relation is invari-ant with respect to the offset length D that, as shown inFig. 1(b), measures the distance from the beginning of thegrating region to the first symmetry axis of the periodicregion. However, D affects the complex phase of the Fou-rier coefficients anm and bnm ; hence Ep and Hp (andtherefore all the fields Ej and Hj) at the input (z 5 0)and output (z 5 L) boundaries of the coupler are func-tions of D.If we now consider Eqs. (2) at all boundaries except
those of the grating layer, i.e., at xj 5 0 for all j Þ p2 1 and p, the continuity conditions for Ej and Hj re-quire that
Ejfj 1 gj 5 fj11 1 Ej11gj11 , (9a)
Yj~Ejfj 2 gj! 5 Yj11~fj11 2 Ej11gj11!, (9b)
with
f0 5 gJ11 5 0. (9c)
Here fj and gj are column vectors with elements $fj%n5 fjn and $gj%n 5 gjn , respectively, and 0 is a null vector,while Yj and Ej are diagonal matrices with elements$Yj%nr 5 yjndnr and $Ej%nr 5 exp(ikjntj)dnr , where dnr isthe Kronecker delta function. The notation in Eqs. (9)using bold capital letters for square matrices and boldlowercase letters for column vectors will be employed con-sistently henceforth. At the grating boundaries Eqs. (2)and (7) analogously yield
S. Zhang and T. Tamir Vol. 13, No. 12 /December 1996 /J. Opt. Soc. Am. A 2405
Ep21fp21 1 gp 2 1 5 A~fp 1 Epgp!, (10a)
Yp21~Ep21fj 2 gp21! 5 BYp~fp 2 Epgp!, (10b)
A~Epfp 1 gp! 5 fj11 1 Ej11gj11 , (10c)
BYp~Epfp 2 gp! 5 Yp11~fp11 2 Ep11gp11!,(10d)
where A and B are square matrices with elements$A%nm 5 anm and $B%nm 5 bnm .By applying a systematic transmission-line analysis12
or resorting to conventional matrix operations, we can useEqs. (9) and (10) to substitute out all but one of the vec-tors fj and gj . As an example, we may single out the setg0 of field amplitudes g0n at the substrate boundary toserve as that special vector. For the guided waves of in-terest here, we then find from Eqs. (9) and (10) that
Mg0 5 0, (11)
whereM is a square matrix whose elements involve func-tions in which the only unknown quantity is the propaga-tion factor k in Eq. (3). If we are to obtain nontrivial val-ues of g0n, the determinant uMu of M must satisfy
uMu 5 0. (12)
The solutions kn 5 bn 1 ian (n 5 1, 2, . . .) for the un-known k in Eq. (12) are thus associated with the eigen-values of M, whose corresponding eigenvectors g0n pro-vide the field amplitudes at the substrate boundary.The reader is referred to other papers12,15 for details
concerning the derivation and the solution of Eqs. (11)and (12). We recall here only that the complex wavenumbers kn form an infinite discrete set, but only two ofthem are usually significant in grating-assisted couplers,as discussed in Section 3. For any guided wave identifiedby a specific kn the corresponding amplitudes g 0n
(n) aregiven by Eq. (11) after a suitable normalization of one ofthem, e.g., g00
(n) 5 1. Once a vector g0n is thus obtained,all the other vectors fjn and gjn can be derived from Eqs.(9) and (10). The fields Ej and Hj of the pertinent guidedwave can then be determined from Eqs. (1) and (7) at anypoint (x, z) inside the periodic interval 0 < z < L.
3. DISPERSION CHARACTERISTICS OF THEGUIDED MODESWhen one uses coupled-wave methods to analyze a grat-ing coupler, it is necessary first to define and find basic(uncoupled) modes that propagate along a coupler con-figuration in which the grating layer has been either re-moved or replaced by a homogeneous medium.3–5 Thegrating is thereafter introduced as a coupling elementthat transforms the basic modes into the so-called coupledones. However, these coupled modes often fail to satisfyorthogonality relations and boundary conditions in the ac-tual periodic structure. By contrast, the modal approachoutlined in Section 2 uses rigorous solutions of the perti-nent boundary-value problem for the actual structure,which includes the periodic layer. This approach thusavoids approximate coupled modes and does not involvebasic modes that are valid solutions only in the absence ofthe grating. In nonperiodic couplers exact modal solu-tions are known as compound modes4 or supermodes,16
but, to the best of our knowledge, rigorous modes havebeen invoked only recently for the description of the op-eration of grating-assisted couplers.11,13 For clarity, weshall therefore use the term rigorous modes (or simplymodes) for exact solutions such as those in Section 2, todistinguish them from coupled modes, for which thequalifying term coupled will always be included.Although rigorous modes are derived directly for the
periodic region (with the actual grating included), weshall nevertheless refer here to a basic (nonperiodic)structure in order to contrast the exact modal approach tothat using coupled modes. For this purpose we charac-terize the periodic layer shown in Fig. 1(b) by
e~z ! 5 ep 1 dp~z !, (13)
where p(z) describes the periodic variation of the dielec-tric constant around its average value ep . In Fig. 1(b)p(z) alternates between only two values, ep1 and ep2, inwhich case ep 5 (ep1L1 1 ep2L2)/L, but other periodicvariations will not affect the qualitative aspects of our re-sults. Here d is a convenient parameter whose range0 < d < 1 provides a measure of the periodic change.Thus d 5 1 corresponds to the actual geometry, and theimpact of the grating layer diminishes as d gradually de-creases. We shall therefore use the limit d 5 0 to definethe basic nonperiodic structure.The (super)modes of the basic (d 5 0) structure vary
as exp(iko Nz), similarly to modes guided by a single di-electric slab. However, because two guiding layers arenow involved, the supermodes appear in pairs having re-fractive indices N that are nearly equal. Consistent withcurrent practice, we assume that the fields in only onepair propagate and that all others are evanescent. If allmedia involved are lossless, the effective refractive indi-ces N of the two propagating modes are given by real val-ues Nev and Nod . Their respective fields, Eev and Eod ,exhibit amplitude peaks in both the bottom and topwaveguiding layers, one of those peaks being usuallymuch higher than the other, as shown in Fig. 2(a). Thefield Eod changes sign at x/l ' 0.2 and therefore exhibitsa null, but Eev undergoes no sign change. In this contextwe recall that Eev and Eod assume the even (symmetric)and odd (antisymmetric) forms shown in Fig. 2(b) for sym-metric coupler configurations.4,16 Of course, such fieldsymmetries are absent in nonsymmetric geometries, butthe subscripts ev and od then indicate that the field am-plitudes exhibit even and odd numbers of nulls, respec-tively.To introduce periodicity progressively in the grating
layer, we can increase the parameter d in Eq. (13) fromzero to unity. The basic modes then gradually evolveinto a pair of different (grating) modes whose final (d5 1) propagation factors kev and kod can be obtained fromEq. (12) as analytical continuations of koNev and koNod ,respectively. Unless the ratio L/l is very small, bothkev 5 bev 1 iaev and kod 5 bod 1 iaod are generally com-plex, i.e., aevÞ0 and aodÞ0 even if all media are lossless.This occurs as a result of radiation (leakage) losses causedby grating diffraction, so that the modes are of the leakyvariety.14,17–19 For practical dielectric gratings it turnsout that bev /ko and bod /ko are close in value to Nev and
2406 J. Opt. Soc. Am. A/Vol. 13, No. 12 /December 1996 S. Zhang and T. Tamir
Nod , respectively, whereas both uaev /kou and uaod /kou areof the order of 1023 or smaller.Typical dispersion curves are shown in Fig. 3, where
the modes in the basic (d 5 0) structure are representedby the nonintersecting dashed lines marked Nev and Nod .The periodicity interval L can then be chosen such that,at some value lph of l, it satisfies
D 5 Nev 2 Nod 5 lph /L. (14)
Equation (14) is regarded as a phase-match condition incoupled-mode theory because the dashed lines for Nev andNod intersect the dotted lines for Nod 1 D and Nev 2 D,respectively, thus producing phase synchronism betweenNev (or Nod) and the 11 (or 21) space harmonic of Nod (orNev). Because higher-order (n Þ 0) space harmonics donot exist in the basic (d 5 0) structure, the dotted linesrefer to dispersion curves of such harmonics only as limitcases for d → 0. We also recall that, if periodicity is in-troduced, Floquet’s theorem20 stipulates that all n5 0, 61, 62, . . . harmonics appear, but only thoseneeded for clarity are retained in Fig. 3. As d increases
Fig. 2. TE-mode fields in structures having two guiding lay-ers: (a) asymmetric case corresponding to the region 0 < z< L in Fig. 1(a) but with the grating replaced by a homogeneouslayer with permittivity ep 5 (e2 1 e4)/2 5 9.62, (b) symmetriccase having t1 5 t3 5 0.5 mm, t2 5 0.8 mm, e0 5 e2 5 e45 3.02, and e1 5 e3 5 3.22. The locations of interface bound-aries are indicated by dashed vertical lines.
from zero to unity, the lines for Nev and Nod and theirharmonics evolve into nonintersecting dispersion curves,as illustrated in Fig. 3 by the solid curves br /ko 1 nDand b l /ko 1 nD, which are located in pairs to the rightand the left of each other, respectively. However, eachsolid (d 5 1) dispersion curve evolves from two differentbasic (d → 0) harmonic curves, e.g., the upper half of thebr /ko curve shown in Fig. 3 evolves from points of thedashed Nev line above the intersection, whereas the lowerhalf of the br /ko curve evolves from points of the dottedNod 1 D line below that intersection.To illustrate the above considerations, in Fig. 4 we
show the variation of kr 5 br 1 iar and k l 5 b l 1 ia l forthe coupler in Fig. 1. Specifically, Fig. 4(b) reveals thatarl and all are reasonably small over wide ranges of l, asexpected, but the corresponding leakage losses may seri-ously affect performance. Furthermore, al undergoesvery sharp variations at several locations in Fig. 4(b).These correspond to rapid changes that occur over narrowbands centered at values of l that satisfy Bragg condi-tions bL 5 qp, i.e., b/ko 5 ql/2L, where q 5 1, 2, . . .and b now stands for either br or bl . However, as cou-plers are usually operated so that Bragg conditions areavoided, we shall not pursue these aspects here and referinterested readers to other sources.12
Coupled-mode methods may be applied to yield disper-sion curves whose form looks similar to that shown in Fig.4(a). In that context the (phase-match) intersectionpoint at lph plays an important role. Coupled-modetheory places this point at the center of the waist betweenthe two solid curves for br and bl in Fig. 4(a). However,rigorous modal analysis reveals that the minimum valueof the ubr 2 b lu gap occurs at some l 5 lgap , which isgenerally unequal to lph . The difference between lphand lgap is often small, as also happens in Fig. 4(a), butthis is not always the case. As a contrary example, Fig.5(a) shows a coupler as in Fig. 1(a), except that now thegrating is above rather than below the upper guiding
S. Zhang and T. Tamir Vol. 13, No. 12 /December 1996 /J. Opt. Soc. Am. A 2407
layer. The accurate dispersion curves in Fig. 5(b) indi-cate that the narrow waist between the br and bl curvesshifts away and to the right of the lph intersection point,which disagrees markedly with the behavior predicted bycoupled-mode analysis. Specifically, the correct minimalubr 2 b lu gap occurs in this case at a wavelength lgap thatis considerably smaller than lph .Although it appears to be inconsistent with coupled-
mode theory, the exact results shown in Fig. 5(b) complyfully with the arguments describing the analytical evolu-tion of dispersion curves for rigorous modes. Specifically,the curves for br /ko and b l /ko are viewed in Fig. 5 asevolving from the lines Nev and Nod of a basic structurefor which Eq. (13) prescribes ep 5 5.62. However, calcu-lations readily show that the same br /ko and b l /kocurves can evolve from a different set of Nev and Nod lineswhose corresponding value of lph is very close to lgap ifep 5 6.65 is assumed. This happens because the modalapproach yields results for the actual periodic structuredirectly without necessarily relating them to a nonperi-
Fig. 4. Dispersion curves for the first two propagating TEmodes in the grating region of Fig. 1: (a) variation of b/ko ver-sus L/l, (b) variation of al versus l.
odic basic configuration. By contrast, coupled-wavemethods require an a priori choice of a basic structure,which leads to dispersion curves whose accuracy may de-pend strongly on the value assumed for ep . Figure 5 sim-ply illustrates a case for which a coupled-mode analysismay be inaccurate if its basic structure was defined by Eq.(13) with d 5 0.
4. BEHAVIOR OF THE MODAL FIELDS ANDPOWER CONVERSIONThe fields belonging to any specific mode in the gratingregion can be accurately determined by the procedure de-scribed in Section 2. Thus, for the mode identified bykr 5 br 1 iar in Fig. 4, the variation of its electric fieldEr 5 Eyr(x, z) is displayed in Fig. 6 for three differentwavelengths. Specifically, Fig. 6(a) shows that the mag-nitude uEru changes only slightly over the periodic inter-val 0 < z < L if l , lgap , i.e., if the operating point onthe br curve in Fig. 4(a) is well above the narrow waistbetween the br and bl curves. Moreover, the shape ofuEru is then very similar to that of uEevu in Fig. 2(a). Thishappens because, for values of br well above lgap as inFig. 4(a), the br /ko curve lies close to that for Nev but farfrom that for Nod 1 D, so that the characteristics of Nevare dominant. (We identify the n 5 0 harmonic of both
Fig. 5. Dispersion behavior of the first two propagating TEmodes in the grating region of a coupler as in Fig. 1, except thatthe grating is on top of the upper guiding layer, i.e., ep 5 (e31 e5)/2 5 (3.22 1 1.02)/2 5 5.62: (a) geometry of the coupler,(b) variation of b/ko versus L/l.
2408 J. Opt. Soc. Am. A/Vol. 13, No. 12 /December 1996 S. Zhang and T. Tamir
Fig. 6. Variation of Er(x, z) for the mode given by kr in Fig. 4 and D 5 0. In (c) the two regions 0 < z/L < 0.5 and 0.5< z/L < 1 are separated for clarity.
br and bl in Fig. 3 by the right-hand pair of curves, whichare closer to Nev than to Nod . This choice is preferredhere because the fundamental mode in the basic structureis usually Nev rather than Nod . However, any other pairof dispersion curves on a Brillouin diagram may be se-lected for identification of the n 5 0 harmonic.)For a value of br well below lgap Fig. 6(b) shows that
uEru also changes only slightly, but its shape resemblesclosely that of uEodu in Fig. 2(a). This occurs because nowbr /ko is close to Nod 1 D but far from Nev , so that Nodcharacteristics dominate. By contrast, Fig. 6(c) showsthat uEru varies strongly with z if the point for br is at l5 lgap . Specifically, uEru starts at z 5 0 with a shapehaving two pronounced peaks in an even arrangement(with a minimum in between), which gradually changesinto an odd form at z 5 L/2 (with a null in between) andthen back to the even shape at z 5 L. Although puz-zling, the field variation in Fig. 6(c) incorporates charac-teristics of both Nev and Nod modes because it is associ-ated with a point of br located at approximately the samedistance from the Nev and Nod 1 D curves.
To clarify the wave behavior shown in Fig. 6(c), wehave examined the space-harmonics composition of themodal fields. For typical couplers as in Fig. 1, we havefound that the dominant harmonics of the kr mode aregiven by n 5 0 and 21 and that all the other harmonicshave much smaller magnitudes. However, the field ofthe n 5 0 harmonic varies like Eev , whereas that of then 5 21 harmonic varies like Eod . Because all kn dependon l, we then takeWn(k, x) 5 Ann(l)wnn(l, x) in Eq. (1)and approximate the field associated with kr by
Er~l! 5 Er~l; x, z ! ' Ar0~l!wev~x !exp~ikrz !
1 Ar,21~l!wod~x !exp@i~kr 2 2p/L!z#,
(15)
where wev and wod identify, respectively, the variationsalong x of Eev and Eod shown in Fig. 2(a), subject to thenormalization
E2`
`
uwev~x !u2 dx 5 E2`
`
uwod~x !u2 dx 5 1. (16)
S. Zhang and T. Tamir Vol. 13, No. 12 /December 1996 /J. Opt. Soc. Am. A 2409
Both wev and wod change slowly with l, so that their de-pendence on x only is retained in Eq. (15). However, theamplitudes Ar0 and Ar,21 vary rapidly with l; specifically,the ratio uAr0 /Ar,21u is generally very large for l! lgap , decreases to unity at l 5 lgap , and then becomesvery small for l @ lgap . When inserted in Eq. (15), thisvariation easily explains the almost constant behaviorshown in Figs. 6(a) and 6(b).For the curves of Fig. 6(c) we take into account that
Ar0 ' Ar,21 at l 5 lgap , in which case Eq. (15) becomes
Er~lgap! ' Ar0~lgap!@wev~x !exp~ipz/L!
1 wod~x !exp~2ipz/L!#exp@i~kr 2 p/L!z#.
(17)
As shown in Fig. 2(a), the principal peak of Eev is in thebottom guiding layer, whereas that of Eod is in the topguiding layer. Hence relation (17) implies that, at l5 lgap , the functions wod and wev combine to yield thedouble-peaked shape of Er in Fig. 6(c), which changeswith z from an almost symmetric (even) to an almost an-tisymmetric (odd) form, and vice versa, over a beat length
Fig. 7. Transverse TE-mode variation of Er , El , and Es fields inthe grating region at z 5 0 and B in Fig. 1(a) with D 5 0 andl 5 lgap 5 1.49775 mm. The 3 signs outline the fields Eod andEev (for d 5 0) as in Fig. 2(a). Layer boundaries are indicatedby vertical lines, which are solid for the grating and dashed forthe other interfaces.
b 5 L/2. We also observe that the even and odd shapesat z 5 0, L/2, and L in Fig. 6(c) resemble strongly thoseshown in Fig. 2(b) for supermodes in nonperiodic direc-tional couplers. However, supermodes retain their indi-vidual shape along z, whereas the modal field Er variesperiodically.The modal field associated with kl can analogously be
but now uAl0 /Al,21u is very small for l ! lgap and verylarge for l @ lgap . We then find that Al0 ' 2Al,21 atl 5 lgap , in which case relation (18) yields
El~lgap! ' Al0~lgap!@wev~x !exp~ipz/L!
2 wod~x !exp~2ipz/L!#exp@i~k l 2 p/L!z#.
(19)
A comparison of relations (17) and (19) reveals that, ex-cept for a shift of L/2, the variations of Er and El with zare almost the same at l 5 lgap . However, unless thebasic structure is chosen so that lph 5 lgap , Er and Elare generally quite different at l 5 lph .We thus find that, at z 5 nL/2, the modes in grating-
assisted couplers operating at l 5 lgap simulate even orodd supermodes in symmetric nonperiodic layered cou-plers (also known as synchronous directionalcouplers).4,16 This implies that those two types of cou-plers have analogous power-transfer mechanisms. Toverify such a possibility, we take Ar0 5 2Al0 5 A0 in afield summation Es 5 Er 1 El , which yields
Es~lgap! 5 Er~lgap! 1 El~lgap!
' 2A0@wod~x !exp~2ipz/L!cos~pz/2B !
1 iwev~x !exp~ipz/L!sin~pz/2B !#
3 exp@i~kav 2 p/L!z#, (20)
where
kav 5 ~kr 1 k l!/2, (21a)
B 5 p/~br 2 b l!. (21b)
Hence, for l 5 lgap , Eq. (20) implies that the total fieldEs at z 5 0, 2B, 4B, . . . is given by wod , in which casethe power flux is concentrated in the top guide. Simi-larly, Es at z 5 B, 3B, 5B, . . . is given by wev , and thepower flux is then concentrated in the bottom guide. Ob-viously, B is the beat length along z over which the en-ergy of Es alternates between those two guides. Super-imposed on this major variation are small fluctuationsover periods L that are due to the terms exp(6ipz/L) inEq. (20). We therefore conclude that, except for those mi-nor fluctuations, Es does indeed behave like the combinedfield of two supermodes in synchronous (nonperiodic) di-rectional couplers.We recall that the above behavior was obtained by the
retention only of the n 5 0 and 21 harmonics for cou-plers of the type in Fig. 1. On the other hand, we foundthat higher-order harmonics may be neglected also inother grating geometries of practical importance, i.e.,
2410 J. Opt. Soc. Am. A/Vol. 13, No. 12 /December 1996 S. Zhang and T. Tamir
those involving values of ej that range from unity to ap-proximately 10. Otherwise, e.g., if metallic media are in-volved, it may be necessary to include higher-order har-monics. However, we expect that the presence of stronghigher-order harmonics will affect only the field shapesrather than the overall power-transfer process.To verify rigorously this power-transfer mechanism, we
have calculated Es by using the exact expressions devel-oped in Section 2 rather than the approximations leadingto Eq. (20). Thus, for the case in Figs. 1, 4, and 6, theaccurate results are given in Fig. 7 for z 5 0 and B. Al-though the Er and El curves do not exhibit the full sym-metries of Eev and Eod in Fig. 2(b), the Es curves clearlyshow that power is effectively transferred from one guid-ing layer to the other. To emphasize this effect, we use 3markers to identify points of the modal field profiles Eevand Eod of Fig. 2(a). These points indicate that the am-plitude of Es is almost identical to that of Eev at z 5 0and to that of Eod at z 5 B. Hence power transfer ingrating-assisted couplers appears to be as effective as innonperiodic directional couplers. However, as discussedin Section 5, other factors may adversely affect the effi-ciency of grating-assisted couplers.
5. EFFICIENCY OF POWER TRANSFERIn actual couplers the field in the grating region is gener-ally given by Es 5 CrEr 1 ClEl , where, in contrast tothe assumption made in Eq. (20), the excitation coeffi-cients Cr and Cl are generally not equal to unity.Specifically, the values of Cr and Cl are dictated by theshape of the incident field and by the boundary conditionsat the input (z 5 0) and output (z 5 L) terminals of thegrating. For typical couplers as in Fig. 1(a), only one ofthe two j 5 2 and 4 guiding layers extends in each of theregions outside the grating, in which case only a singlemode propagates in those exterior regions. We thereforedenote the fields of these modes by Et at z , 0 and Eb atz . L to identify their being guided by the top and thebottom layer, respectively. Obviously, Et and Eb aregenerally quite different from the supermode fields Eevand Eod , which appear when both guiding layers arepresent. As a field Et (or Eb) is incident onto the coupler,the abrupt terminations of the guiding layers at the ter-minal planes z 5 0 and L act as strong boundary discon-tinuities that cause the incoming power to be unevenly di-vided between the modal fields Er and El inside thegrating region. Hence the total field in the grating canbe quite different from Es given in Eq. (20) and Fig. 7, sothat complete power conversion may not occur.To evaluate these aspects, we first consider a grating
coupler whose guiding layers extend well beyond the grat-ing region, as shown in Fig. 8(a). In this case the fieldssupported outside the grating region are given by Eev orEod , and the boundary discontinuities at its input andoutput terminals are weak rather than strong. For afield Eod incident from the left we then have
E in~x, 0! 5 wod~x ! ' CrEr~x, 0! 1 ClEl~x, 0!
5 Es~x, 0!, (22)
where l 5 lgap is implied but omitted henceforth in allthe terms E(l; x, z). Equation (22) neglects wave scat-tering that is due to reflections and the possible excitationof other (mostly evanescent) modes at z 5 0. However,by carrying out energy balance calculations for practicalsituations, we have ascertained that those effects extractonly a small amount of power (approximately 3% or less)from the incident field.The fields Eev and Eod , as well as Er and El , are given
rigorously by Eqs. (1)–(5), (7), and (8) and relation (16).Also, the discussion following Eq. (8) has established thatthe functional variations of Er and El depend on the off-set distance D shown in Figs. 1 and 8(a). By multiplyingboth sides of Eq. (22) with Er* or El* (the asterisk denotingthe complex conjugate) and integrating the result over x,we obtain Cr and Cl as overlap integrals in the form
Cr 5 Cr~D! >E
2`
`
wod~x !Er* ~x, 0!dx
E2`
`
uEr~x, 0!u2 dx, (23a)
Cl 5 Cl~D! >E
2`
`
wod~x !El* ~x, 0!dx
E2`
`
uEl~x, 0!u2 dx. (23b)
Fig. 8. Power transfer in a coupler with weak terminal discon-tinuities. (a) Geometry and power flow in a coupler having thesame parameters inside 0 , z , L as those in Fig. 1(a); outsidethat region the grating layer continues as a homogeneous layerwith permittivity ep 5 (e2 1 e4)/2 5 9.62. (b) Power ratios ver-sus L for l 5 lgap and lph . The curves in (b) are shown for D5 0, but they are not discernibly different if D Þ 0.
S. Zhang and T. Tamir Vol. 13, No. 12 /December 1996 /J. Opt. Soc. Am. A 2411
At the output (z 5 L) plane the field is then given by
Es~x, L ! 5 CrEr~x, L ! 1 ClEl~x, L ! ' Cevwev~x !
1 Codwod~x ! 5 Eout~x; D, L !, (24)
where wave scattering at the z 5 L plane is also disre-garded, and
Cev 5 Cev~D, L ! 5 E2`
`
Es~x, L !wev* ~x !dx, (25a)
Cod 5 Cev~D, L ! 5 E2`
`
Es~x, L !wod* ~x !dx. (25b)
To discuss power transfer, we denote the powers carriedby the modal fields Eev , Eod , Er , and El by Pev , Pod , Pr ,and Pl , respectively. When TE-mode power is trans-ferred from an input field Eod into an output field Eev , thecoupler has a conversion efficiency
hod→ev~D, L ! 5Pev~L !
Pod~0 !5
Nev E2`
`
uCevwev~x !u2 dx
Nod E2`
`
uwod~x !u2 dx
5Nev
NoduCevu2, (26)
where the last equality follows from Eq. (16). The con-version of an input field Eev into a field Eod at the outputcan be expressed by an analogous efficiency hev→od . Con-version efficiencies for TM modes can similarly be ob-tained, but magnetic Hj fields must then be used insteadof electric Ej components.For the coupler in Fig. 8(a) the conversion efficiency
given by Eqs. (23)–(26) is plotted in Fig. 8(b). We thusfind that the power Pev at the output changes from zero toa maximum, or vice versa, over distances L equal to inte-gral values of B ' 0.7 mm, as expected. The maxima de-crease slightly with L because of radiation leakage, whichis accounted for by the exponential terms having complexvalues of kr and kl in Eq. (24). More importantly, thepeaks in Fig. 8(b) are lower if the operating wavelength islph (dotted curve) rather than lgap (solid curve). Theanalytical formulation does not explicitly provide the ex-act value of l for which those peaks are maximized, butextensive calculations have determined that this occursat some l that is numerically very close (and probablyidentical) to lgap . This result is consistent with physicalintuition because, at l 5 lgap , the fields Er and El in Fig.7 for grating couplers show greatest similarity with Eevand Eod in Fig. 2(b) for symmetric nonperiodic couplers.We therefore assume henceforth that the power peaks aremaximized at a wavelength lgap that, as discussed in Sec-tion 3, is generally different from lph .The power sum Pout 5 Pev 1 Pod at the output is
shown in Fig. 8(b) by a dashed curve, which fluctuatesslightly as its magnitude decreases exponentially with L.This happens because the fields Er and El in the gratingregion have somewhat different decay factors ar and al ,respectively, as can be seen from Fig. 4(b). It is theneasy to verify from Eq. (24) that Pout /P in will generally
decrease in a fluctuating manner. Most importantly, wehave found that, if the offset distance D is changed, thecurves for the new D values are almost indistinguishablefrom those already shown in Fig. 8(b) for D 5 0. This isexpected for boundary discontinuities that are not abrupt.Because our results show that such a behavior also holdsfor other grating configurations, we conclude that the con-version efficiency h (D, L) is effectively independent of D ifthe discontinuities at the input and the output are of theweak type.To examine power transfer in a realistic coupler having
abrupt boundaries as in Fig. 1(a), we recall that the topguide (at z , 0) supports a single guided field Et with ef-fective refractive index Nt Þ Nod and the bottom guide (atz . L) similarly supports a single modal field Eb with in-dex Nb Þ Nev . Power conversion can then be assessedby the use of the approach given in Eqs. (22)–(26) pro-vided that we replace therein all subscripts od and ev by tand b, respectively. Because the boundary discontinui-ties in Fig. 8(a) are weaker than those in Fig. 1(a), thefields neglected in the latter case may considerably de-grade the accuracy of the modified equations. However,the calculated efficiencies h (D, L) obtained by the neglectof those scattered fields tend to be higher than the correctones. Nevertheless, the properly modified Eqs. (22)–(26)yield significant results in the sense that they determinemaximum achievable efficiencies.With these qualifications the efficiency of the coupler in
Fig. 1(a) to convert incident power Pt in the Et mode intooutgoing power Pb in the Eb mode is shown in Fig. 9.
Fig. 9. Power transfer in the realistic coupler of Fig. 1: (a) ge-ometry showing power flow, (b) variation of efficiency h t→b withL when power Pt is transferred from the top guide to Pb in thebottom guide.
2412 J. Opt. Soc. Am. A/Vol. 13, No. 12 /December 1996 S. Zhang and T. Tamir
The abrupt terminal discontinuities manifest themselvesin two ways: (1) The efficiency curves exhibit peaks hav-ing heights and locations that are now strongly depen-dent on the offset distance D, and (2) substantial fluctua-tions occur within every periodicity length L. Wetherefore conclude that strong discontinuities at the inputand output terminals may appreciably lower the conver-sion efficiency. We also note that, as this paper went topress, a study of such discontinuity effects was reportedby Little,21 who used coupled-mode and variational tech-niques to obtain results that agree very well with thoseobtained here.In principle, one may avoid losses in efficiency by de-
signing D and L to yield peak-power operation, such asthat for D 5 L/4 and L 5 0.825 mm in Fig. 9. However,this would necessitate fabrication tolerances that may bedifficult to implement. A more practical solution is to re-place abrupt discontinuities by gradual transitions thatprovide adiabatic changes in the fields at the grating ter-minals. Possible schemes for implementing such smoothtransitions are shown in Fig. 10. In particular, Fig. 10(a)describes waveguides that taper vertically away from theoutput and input terminals of the grating. As such a ver-tical x-plane tapering may not be easily fabricated, Fig.10(b) shows an alternative scheme in which thewaveguides are tapered in the horizontal y plane. Theseor other transitions providing good field-matching condi-tions should achieve the optimized power-conversion re-sult represented by the solid curve for lgap in Fig. 8(b).
6. CONCLUSIONSWe have examined the field structure and the power-conversion mechanism in grating-assisted couplers by us-ing a rigorous modal approach. Unlike other methods,this approach can address couplers having large gratingheights and strongly coupled waveguides. Our analysisof the modal fields shows that, except for losses that aredue to energy leakage, the process of transferring powerfrom one waveguide to another in grating couplers issimilar to that in nonperiodic directional couplers.
Fig. 10. Schemes for avoiding deterioration in power-conversionefficiency.
We have also assessed the efficiency h of convertingpower incident at the input of a grating-assisted couplerinto a desirable mode at its output. The results revealthat h is highest at a wavelength lgap that is generallydifferent from the phase-match wavelength lph used bycoupled-mode techniques. Furthermore, we have foundthat abrupt geometrical discontinuities at the beginningand the end of the grating region can adversely affect theefficiency h. This efficiency deterioration may be avoidedby judicious design of the input and output terminals ofthe coupler, and tapered transitions that achieve this pur-pose were suggested.
ACKNOWLEDGMENTSThe authors thank G. Griffel for discussions and construc-tive suggestions, and K. C. Ho for helping out with theanalytical derivations. This study was supported by theU.S. National Science Foundation. S. Zhang partici-pated in this research during a leave of absence from theHangzhou Institute of Electronics Engineering, Hang-zhou, China, with partial support from the NationalNatural Science Foundation of China.
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