Effectiveindex method and coupledmode theory for almostperiodic waveguide gratings a comparison - PDF document

Effective-index method and coupled-mode theoryfor almost-periodic waveguide gratings:a comparisonKim A. WinickContradirectional propagation through active, first-order, almost-periodic, corrugated waveguide grat-ings is analyzed by using both coupled-mode theory and a combined effective-index/impedance-matchingmatrix technique. For TE-mode operation, which is near the first-order Bragg wavelength, theequivalence of the two techniques is analytically demonstrated for shallow surface corrugations.Key words: Effective-index method, coupled-mode theory, corrugated waveguide gratings.I. IntroductionThe propagation convenient matrix form and used it to study randomfluctuations in first-order grating filters. RecentlyBjork and Nilsson22used this approach to study theproperties of asymmetric phase-shifted DFB lasers.Basu and Ballantyne's approach is a direct applica-tion of the effective-index method combined withimpedance matching. Each period of the waveguidegrating is divided into thin sections, and the guideheight and material gain are assumed to be constantwithin the section. The section is treated as a three-layer waveguide, and the standard b versus V disper-sion relationships are used to compute the propagat-ing modes that are supported by this guide. A 2 x 2transfer matrix for the section is then derived bymatching the tangential E and H fields (which corre-spond to these modes) at the interfaces betweensections. Finally, the transfer matrix for the completestructure is obtained by multiplying together theindividual transfer matrices.Verly et al.23'24derived the effective-index method inperiodic corrugated gratings directly from Maxwell'swave equation and a local normal mode expansion ofthe field.25In this way they were able to examine theapproximations inherent in the effective-index tech-nique. They demonstrated that it was a relativelyaccurate theory for TE modes and could be modifiedto give correct results for the TM case. Finally, theycombined the effective-index technique with coupled-wave theory for one-dimensional dielectrics. Usingthis combined approach, they obtained results thatwere identical to those obtained by use of the coupled-mode theory for periodic waveguide gratings.We analytically demonstrate by direct computationthat the coupled-mode theory technique is equivalentto the TE-mode effective-index/impedance-matchingmethod. We do this for almost-periodic waveguidegratings, which may have gain. As opposed to Verly'smethod, the coupled-wave theory results for one-dimensional dielectrics are not used in our develop-ment. Contradirectional propagation through an al-most-periodic, corrugated waveguide grating isassumed, as is operation near the first-order Braggwavelength.II. Impedance-Matching Matrix MethodConsider the planar, thin-film waveguide shown inFig. 1. This is a three-layer dielectric guide, consistingof cover, film, and substrate layers, with refractiveindices n, n, and n, respectively. The interfacebetween the film and cover layers has a shallow,surface corrugation, which can be modeled as analmost-periodic square-wave grating. For simplicitywe divide the grating into slabs, numbered 0 throughN + 1 as indicated in Fig. 1. The width of the kth slabis Wk, and its height is hk. Light of free-space wave-length X is confined to the film region by total internalreflection at the film-cover and film-substrate inter-faces (n n nf). It is assumed that the corruga-tion effectively couples only two contradirectional,TE-polarized, guided waves. These TE-polarizedwaves have an electric field component only along thexzync hkflf 'U ', 1 .Wk &#x 000; .@ ' : I O .1 : 2 ' § , " k k+1 N'N+1zIZkT Zk+1ZN+1Fig. 1. Almost-periodic waveguide grating.y direction. We write the y component of the electricfield in the kth slab asE8(x, y, z) = EFk(X, Y, Z) + EBk(X, Y, Z),(1)whereEF(x, y, z) = AkETA(x) exp(-i3z),EBk(x, y, z) = B8ETk(x) exp(i1kZ),(2)(3)and EFk and EBk denote the forward (+z direction) andbackward (-z direction) propagating fields. The timedependence of the fields is assumed to be exp(iwt).ETk(x) and Pk are the transverse-mode profile and thepropagation constant, respectively, for an uncorru-gated guide of height hk.The propagation constant Pk can be computed byusing the standard, three-layer guide, dispersion equa-tion given below'5:Vk(1 -bk)" = tan'[(bk + a)/(l -b )]1/2+ tan'[bk/(l -bk)]12+ v.r, (4)whereVk = -hk(nf -n. Ia = (n 2-n 2)/(nf2-n2),Nh2n 2bk = 2_ 2 2 2 2flf fl827rrPk= yNk,(5)(6)(7)(8)and v is an integer denoting the mode number.Observe that Nk is the effective index of the kth slab.Using Maxwell's equation, we obtainV x E = -iwuH,(9)where u is the permeability of free space. The xcomponent of the magnetic field in the kth slab,758 APPLIED OPTICS / Vol. 31, No. 6 / 20 February 1992 H,(x, y, z), is given byHk(x, y, z) = HFk(X Y, Z) + HBk(X Y z),(10)the end. For example, the reflection coefficient r(X), asdefined by() EBO(X, Y, Zi-)En(x, y, z,-)where(20)HFk(x, Y, Z) = -Ek(x, Y, Z) (11)HBk(x, Y, Z) =P EBk(X, Y, Z) (12)Let Zk- and zk+ denote the values of z just to the left ofand to the right of Zk, respectively, where z, lies on theboundary between the (k -1)th and kth slabs. Theelectromagnetic boundary conditions require that E.and Hk be continuous across the interface betweenthe (k -1)th and kth slabs. Therefore, it follows fromEqs. (1), (8), and (10)-(12) that[EFkl(x, y, Zk8) 1 EF Y, Zk')'k-1 I = 1 IE -s (X, y, Zk ), E (X, y, Zk+)where the matrix k is given by'L 1 1=-Nk Nk.It also follows immediately from Eqs. (2) and (3) thatpropagation across the kth slab can be written as[EFk(x, y, Zk )] [EFk(x, y, z4+, )1=- I , (15)EBk(x, y, Zk) J [EBk(x, Y, Zk+ )with EBN(x, y, ZN+1-) set to 0, can be calculated directlyfrom the ik matrices by using Eq. (19).Iil. Evaluation of orCombining Eqs. (8), (14), (16), and (18) yields2,BN) exp[wk(iPA8-gk)] 21 expl-w(ik -gk)]2, exp[W(i3, -g)] 1 -2J exp[-w(ipk -g)](21)(13) whereAN = Pk-l -k, (22)Therefore, the transfer matrix for a single period of(14)the surface corrugation is given bya bAffkll = d k = 1, 3,5. ...wherea = 1 2 1 -eXP[i(YkWk + 'Y+lWk+l)]2 'kJ 2138~(23)where the matrix W. is given by0 X~~~~~= exp [wk (i ANk -g8)] ex w 1~8-(6and gk denotes the material gain, if any, in the kthslab. Combining Eqs. (13) and (15) yieldsEFk-l(X yY,Zk)] [EFk(X Y, Zk+l )]= (- k* (17)LEsk-(X, Y, Zk[EBk(X, Y, Zk+),Defining the one period transfer matrix x/k as= ~k-l k (18)and using Eq. (17) repeatedly, we can writerEF(x, Y. Z1l [N2 EFN(X, Y, ZN+1 1I 1 Y, = _ -l4'2k-1 E (19)LEsO(x, Y, Z1 ) \ IEBN(x, Y. ZN+1)J,where we have assumed that N is even. Thus the xikmatrices completely characterize the propagationthrough the waveguide grating. Their product relatesthe fields at the beginning of the grating to those at428 2k ePxp-i (YkWk -Yk+lWk+l)], (24)b = (1 2 213 exp[i(ykwk -k+lWk+l)+ (1i- Ai' As exp[-i(,y+lwk+l + Ykwk)], (25)2138 2Pk-1c = _1- 213 ) 21 , exp[i(YkWk + Yk+lWk+l)]+ (1 A + exp-i(YkWk -Yk+lWk+l)], (26)213k, 2 xp-d = 1 2 1) 1(- 2k ) exp-i(YkWk + Yk+lWk+l)]+ Ak 213 exp[i(ykWk -y+lWk+l)], (27)k = k + igk-(28)The expressions for a, b, c, and d given above can besimplified. First, we observe that for an almost-periodic grating,WkWk+l,(29)20 February 1992 / Vol. 31, No. 6 / APPLIED OPTICS 759
1Ak = (30) where(31) P1ay(Z) = 2 for Zk Z Zk+2, k = 1, 3, 5, ....23+3+ ISecond, we assume that the depth of the surfacecorrugation is small compared with the averagewaveguide height so thatk Z = + k+1 7r=82 2wkfor Zk Z Zk+2, k = 1, 3,5, ....I A 11k-1(32)and that the gain per slab section is small so thatg9wk 1. (33)Finally the deviation a, from the local first-orderBragg condition is defined by28 = ( + Tr+)-- k = 1, 3, 5, ....w 81) -(34)and we assume that the Bragg condition is nearlysatisfied, or equivalently,(44)Combining relations (29), (34), and (40) and Eqs.(41)-(44) (along with the fact that ak-1 = a, anda,~ = k [ 3k +, for an almost-periodic grating) yieldsR(zk-) exp[2wk(ibk -g8) R(Zk,+2)rexp i ) SZ )(45)S1k- 8exp / k ('+-Pk (_ Wk )+ exp[-2w8(i8-g)] S(Zk+2 ). (46)Without loss of generality we set z, (the z coordinateat the beginning of the grating) to zero; thenI 8,w81.(35)Combining Eqs. (24)-(35) now yields the followingfirst-order expressions for a, b, c, and d:a = -exp[2w8(i&k -g)],AkArkc --,(36)k-1Zk = I Wj.j=1(47)The grating is assumed to be almost periodic. There-foreI r -1 for k even,exp -i -zk = exp[±i(k -1)'r] =(37)\ W / 1+1 for k odd.(48)Combining approximations (45), (46), and (48) yields(38) to first orderd = -exp[-2wk(ik -gk)]. (39) [R(Z -A1k3In deriving relations (36)-(39) we used the fact that Pk Je =1+q forqI 1and neglected terms of the form (A[II)1(8wYP2(gWy3when p1 + p2 + p3 &#x 000; 1. This degree of approxima-tion is consistent with the conditions given by inequal-ities (32), (33), and (35). Combining Eqs. (17), (18),and (23) and approximations (36)-(39) yieldsIEFkl(X, Y. zi )1 [-exp[2w(i88-gk)]I -(X, Y, Z LE ,, ,(XI y, zki ) -13__Ok13k-exp[-2wk(iak gk)].EF+,(X y, Zk+2, )X .~+ 1 (XI Y, Zk +2 ),Ak l [R(Zk+2-)exp[-2wk(i5k -g,)]* (L+2 )(49)for k odd.IV. Coupled-Mode TheoryBased on a coupled-mode theory analysis, R(z) andS(z), as defined by Eqs. (41) and (42), are solutions tothe following pair of differential equations5:R '(z) + [i (Z) -g(z)]R(z) = -K(Z)S(Z),S'(z) -[ib(Z) -g()]S(Z) = -K(Z)R(Z),(40)We now define the functions R (z) and S(z) as follows:EFY(X, Y, z) = R(z) exp[i8(z)z] exP[-iav(z)z], (41)EBY(x, y, z) = S(z) exp[-i8(z)z] exp[iav(z)z], (42)(50)(51)where the prime denotes differentiation with respectto z and K(Z) is the coupling coefficient. For TE-modepropagation in the guide, the coupling coefficient K(Z)for an almost-periodic, square-wave, surface corruga-tion (see Fig. 1) is given by"2 hk+, -hk nf NkK8= K(Z) = ~ hef N,for zk Z 42, k = 1, 3, 5,..., (52)760 APPLIED OPTICS / Vol. 31, No. 6 / 20 February 1992(43)Ark -Akl,9k =~ gk+l- where hff8is the effective height of the guide and isapproximately equal to"V8+ bj"' + (b, + a)"'heff 2r ' k = 1, 3, 5, .... (53)-(72 2)1/2(f _ 8nIt is assumed in approximation (52) that the corruga-tion depth is shallow, and therefore the electric fieldof the uncorrugated waveguide mode is taken to beconstant over the corrugation depth. For operationnear the first-order Bragg wavelength, Eqs. (8) and(34) and inequalities (32) and (35) can be combined toyield2-N,,2Wk = 1 (54)Thus it follows from approximation (52) and approxi-mation (54) thatKk2Wk= hef -Nh ' k = 1, 3,5,....It is now shown thatKk2Wk = , k = 1, 3,5,....(55)It now follows from Eqs. (8) and (61) and approxima-tion (53) thatdo dN nf2 -N dh13_ N N 2 __ .f N N2 h~~~eff(62)Combining approximation (55) and Eq. (62) yields thedesired resultKh2Wk = 82,, k = 1, 3, 5, ....Pk,(56)Since R (z) and S(z) change little over the distanceof one grating period, we can write(63)R(zk ) = R(z+,2) -R (Zk,+2)2Wk,S(Zk ) = S(Zk,+2) -S'(Zk,+2)2Wk.Combining Eqs. (50) and (51) and approximations(63) and (64) yieldsR(zk-) = R(z,+2,) + 2wk(ibk -gk)R(k+,2) + K2WS(Zk+, ), (65)S(zk ) = S(zk+,2 -2wk(ib, -g,)S(zk,+2) + K82wkR(Zk,+,). (66)Recall from inequalities (33) and (35) that g^Wk 1and I 8wk 1. Therefore,(56)exp[±2wk(ibk -g)] = 1 + 2Wk(i8k -gk).(67)(64)We start by differentiating between botl(4) with respect to h. The result isdV 1121db 11[1 -b]"' -2 V (1-b)1' = b + a 2 b+1 -1b(-b) + (b + a) db 1 11-b\"' (:x -+ _ _-(1- b)2 dh 1 b 2 b1 -b1 1 r 1-2 (1- b)"'[b"'2 (b +It follows from Eqs. (5) and (7) thatdV 2Tr 2)1/2,da = (.f' -n82,dN (nf2 -n,')db 2Nnf2 -N21-b= 2 2nf n,Multiplying both sides of Eq. (57) by [(dN/db) and then using Eqs. (58)-(60) yinf2-N'N[V + b1 + (b + a)"12sides of Eq., 1i/21 -b) + b db(1 -b) 2 dhCombining approximations (56) and (65)-(67) yieldsrR(z,,-)| 1 exp[2wk(ib8g,)]I -)= AN3S(Zk-) Pk~~~~~~~3,AN1,P~kexp[-2wk (i8k g8)]x (Z+2 )LS(Zk+2 )AFinally we observe that and approximations (68)L db (49) are identical. Therefore an identical result fora)1 dh- (57) the one-period transfer matix /'kg+1 is derived byusing either the effective-index/impedance-matchingtechnique or coupled-mode theory.Approximations (49) and (68) indicate that theeffective-index method and coupled-mode theory yield(58) nearly identical results for almost-periodic corru-gated waveguides in the vicinity of the first-orderBragg wavelength. Our analysis, however, has been(59) restricted to grating profiles that have a squareshape. In coupled-mode theory an arbitrary gratingprofile is handled by decomposing the profile into its(60) Fourier series components. The fundamental compo-nent has the same grating period as the originalprofile, and the periods of the remaining components1 -b]" (dh) are integer fractions of the fundamental. It is well.eld known in coupled-mode theory that these remainingcomponents can be neglected, since they do notIN. (61) achieve phase matching at the first-order Bragg wave-length. Thus a shallow grating profile of arbitraryshape can be replaced by an equivalent square profile.20 February 1992 / Vol. 31, No. 6 / APPLIED OPTICS 7612 [n 2 _n2il/22'rnr -n The only restriction is that the fundamental Fouriercomponents of the two profiles have the same ampli-tude and period. Therefore the effective-index methodand coupled-mode theory will produce nearly identi-cal results even for shallow grating profiles of arbi-trary shape. The effective-index analysis, however,must be performed on the equivalent square profile asdefined above.V. Recursion MethodsA simple recursive technique is widely used in thin-film design to determine the reflection coefficient of amultilayer dielectric structure in terms of the reflec-tion coefficients of the individual layers.26This recur-sive procedure is often referred to as the Airy summa-tion technique or Rouard's method. In Refs. 17 and18 an Airy-like summation procedure is used toanalyze corrugated waveguides, but no mathematicaljustification for the technique is provided. Below weshow that the recursion method is equivalent to firstorder to a numerical integration of the coupled-modeequations.Using approximation (67), we can write approxima-tions (65) and (66) in the following matrix form:[R(zku) exp[2wk(ibk gk)]S(Zk, -) I 2WkKk2WkKk 1exp[-2wk(i8k -901L(Zk+2 ) XI we d. (69)If we define the reflection coefficient r, of the com-bined k and k + 1 layers asS(Zk ) I= R(Zk ) SUk2-)=0it follows from Eq. (69) thatrk = 2kKk exp[-2Wk(ibk -901(70)(71)combined to yieldPk rk + Pk12 exp[-4wk(ik g,9)]1 + rkPk+2(75)Approximation (75) is the desired recursion relation-ship, and the above analysis indicates that it isequivalent to first order to a numerical solution of thecoupled-mode equations.VI. TM CaseWe have shown that for TE polarization the com-bined effective-index/impedance-matching techniqueyields the same result as the coupled-mode theorydoes. The development shown above can be paralleledfor the TM polarization, and approximation (49) isobtained unaltered. It is known, however, that thecoupling coefficients are different for TE and TMpolarization.'5Therefore the effective-index methodand the coupled-mode theory methods do not yieldthe same results for TM polarization.'4As noted byVerly et al.4this occurs because the effective-indexmethod does not account for the boundary conditionsat the grating-cover plate interface. For TE polariza-tion this is no problem, since the E and H fields arecontinuous across this interface. For TM polariza-tion, however, there is a periodic discontinuity in E,at the interface. This periodic discontinuity gives riseto an additional coupling term, which the effective-index method neglects. It is easy to show that thecoupled-mode theory and the combined effective-index/impedance-matching technique will yield iden-tical results provided that the Ap,, I [,, term in approxi-mation (49) is reduced by the following factor:N."21nf2_ Na2/nc2 + 1Nav'/f + N2/nc2 -1(76)whereXFurthermore, if we let Pk denote the reflection coeffi-cient of the grating as seen when looking in thepositive z direction at position z = Zk- thenS (Zk )|R(Zk) (72)Combining Eqs. (69) and (72) yields[R(Zk ) ] [exp[2wk(ik -gk,)]LPR(Zk) L 2WkKk2WkKk1exp[-2 k(ibk g_,)] X~ +2R (Zk,2 ) 73It immediately follows from approximation (73) that2WkKk + Pk+2 exp[-2Wk,(ik -gk)] (74= exp[2wk(iBk -g)] + Pk+22WkKkEquation (71) and approximation (74) may now be(77)VlI. ExamplesThree corrugated waveguides have been chosen tocompare the effective-index technique with the cou-pled-mode theory. We implemented the effective-index technique, using Eqs. (4)-(7), (14), (16), and(19), as described in Section II. The coupled-modetheory results were found by numerically integratingEqs. (50) and (51), starting at the back of the gratingand moving forward. The integration step size was-0.2% of the grating length, and K(z) and 8(z) wereevaluated by using Eqs. (4)-(8) and (34) and approxi-mations (52) and (53). The results are shown in Figs.2-4, where the reflectivity, EBo (z1j)/EF0 (z,) 2, isplotted versus wavelength. As expected there is excel-lent agreement between the two approaches.The waveguide parameters shown in Figs. 2-4 arespecified by using the notation of Fig. 1. In all threecases the film, substrate, and cover plate refractiveindices are 1.55, 1.5, and 1.0, respectively, and the762 APPLIED OPTICS / Vol. 31, No. 6 / 20 February 1992 Z' 0.6-0.4-cc0.2-0.01.48 1.49 1.50 1.51 1.52Wavelength (yi)Fig. 2. Periodic grating n, = 1.0, nf = 1.55, n8= 1.5, w = 0.247pam, h, = [1.5 + 0.15(- 1)] pm, k = 1, 2, ..., 1000 layers}.average film height, (h +k + hJ)/2, is 1.5 pm. Thesevalues correspond to silver ion-exchanged waveguidesthat are made in glass.27At a wavelength of 1.5 plmthe effective index Neff and the effective guide heightheff are found to be 1.51822 and 2.724 pum, respec-tively. Waveguide 1 (see Fig. 2) has a constantcorrugation period equal to 0.494 [im and a constantcorrugation depth, i.e., h,,+ -hk I, of 0.30 ,um. Thedevice is 247 sum long and consists of 500 corrugationperiods. The second waveguide (see Fig. 3) is also 247pm long and consists of 500 corrugation periods. Thecorrugation depth, however, has a raised cosine taper,which yields a maximum depth of 0.30 plm at thecenter of the guide and no depth at either end. Asexpected, the amplitude taper reduces the sidelobelevels of the waveguide filter. The third waveguide(Fig. 4) has a constant corrugation depth of 0.30 pm.The period, however, varies linearly along the lengthof the guide, ranging from 0.2445 plm at the begin-ning to 0.2495 pm at the end. This variation corre-sponds to a 2% linear chirp over a length of 5000.8_ coupled modeeffective index0.6,3~~~'5 0.4-ID0.2-0.0 --1.48 1.49 1.50 1.51 1.52Wavelength (am)Fig. 3. Amplitude-tapered grating (n, = 1.0, nf = 1.55, n,= 1.5,Wk = 0.247 p.m, h, = [1.5 + 0.15(-1) ] 10.5 + 0.5 cos[rr(k -500)/500]} pm, k = 1, 2_ .., 1000 layers).0a)a)IDZcc1.52Wavelength (m)Fig. 4. Frequency-chirped grating n, = 1.0, nf = 1.55, n8= 1.5,Wk = [(-0.0025 + 0.0025k/500) + 0.247] ptm, h, = [1.5 +0.15(-1)'] plm, k = 1, 2_ ., 1000 layers}.corrugation periods. As indicated in Fig. 4, the chirpbroadens the spectral response of the waveguidefilter.Vil. ConclusionsPeriodic and almost-periodic waveguide gratings havefound applications as filters, DFB laser structures,phase-matching elements, grating couplers, pulse com-pressors, and soliton generators. Both coupled-modetheory and a combined effective-index/impedance-matching technique have been used to analyze propa-gation through these devices. The coupled-mode the-ory methods are older and have been applied morewidely. The effective-index/impedance-matching tech-nique, however, is conceptually simpler. In the matrix-based effective-index technique, the grating periodregion is divided into thin-slab sections. The modepropagation constants in each of the slabs are evalu-ated by using the standard b versus V dispersionrelationships for a three-layer waveguide. A two-by-two transfer matrix for each section is then derived bymatching the tangential E and H field components atthe interface between slabs. We have shown by directcomputation that the matrix-based effective-index/impedance-matching technique is equivalent to cou-pled-mode methods for TE polarization.I thank the anonymous reviewers for their helpfulcomments. This work was supported in part byNational Science Foundation grant ECS-8909802.References1. D. C. Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank,"Grating filters for thin-film optical waveguides," Appl. 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