Difference between TE and TM modal gain in amplifying waveguides: analysis and assessment of two perturbation approaches

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1 Optcal and Quantum Electroncs 29 (1997) 263±273 Dfference between TE and TM modal gan n amplfyng wavegudes: analyss and assessment of two perturbaton approaches J. HAES, B. DEMEULENAEE,. BAETS Unversty of Gent-IMEC, Department of Informaton Technology (INTEC), St-Petersneuwstraat 41, 9000 Gent, Belgum D. LENSTA Free Unversty, Department of Physcs and Astronomy, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands T.D. VISSE, H. BLOK Delft Unversty of Technology, Department of Electrcal Engneerng, P.O. Box 5031, 2600 GA Delft, The Netherlands eceved 8 January; accepted 10 June 1996 The commonly used con nement factor-based formula for modal gan n amplfyng wavegudes ± g mod = Gg mat, wth G a con nement factor ± s well establshed and accurate for TE modes. The TM case s rarely, and sometmes erroneously, descrbed n the lterature. Usng a varatonal formulaton the fundamental dfference between TE and TM modal gan s llustrated. An accurate expresson, correct up to rst order, for the TM modal gan s then derved from a known general perturbaton formula. However, as ths does not lead to a true con nement factor formulaton, some approxmatons are ntroduced, leadng to a un ed formulaton of both TE and TM modal gan. A second method to calculate the modal gan, based on the analytcty of the dsperson equaton, s also dscussed. Smulaton and comparson wth modal gan values from a complex mode solver wll nally llustrate the valdty of the dfferent approaches. 1. Introducton One of the most mportant propertes of amplfyng wavegudes, such as lasers or optcal ampl ers, s the modal gan experenced by d erent modes. It has been known for a long tme [1] that for TE slab modes, the modal gan s gven by a con nement factor or llng factor-based expresson. The modal gan can be wrtten as a weghted sum of the con- nement of the domnant TE eld component n each wavegude layer. The weghng coe cents contan as the most mportant factor the materal gan of each layer. Very often ths formula s used as a perturbaton approxmaton of the modal gan n the amplfyng wavegude by usng the con nement factor of the eld n the wavegude wthout gan or loss. Ths avods the need for a complex modal calculaton and facltates the smulaton of the e ects of d erent gan or loss levels n wavegude desgn. 0306±8919 Ó 1997 Chapman & Hall 263

2 As wll be emphaszed n ths paper, TE and TM modal gan behave fundamentally d erently [2]. Ths fact s very often not recognzed n the lterature. In [1] and [3] the dscusson s lmted to TE polarzaton and n [4] and [5] the TE expresson, usng the con nement factor for the domnant TM eld component, s wthout motvaton appled to calculate the TM modal gan. In [6] the modal gan s derved under the assumpton of weak gudance, where the d erence between TE and TM modes vanshes. Only n [7] and n [8] are the dstnctons between both polarzatons made. Furthermore n [8] a second possblty for modal gan calculaton, ncludng both polarzaton states, s suggested, but not appled. There t s remarked that the dsperson equaton for the modal e ectve ndex s an analytc functon of the refractve ndex pro le. We wll assess ths approach and compare the results wth mode solver and perturbaton results. In [9] t s shown that the correct modal gan formula for TM slab modes d ers substantally from the TE expresson. The expresson derved n [9], although exact, s less useful as a startng pont for a perturbaton approach. Vassallo formulates n [8] a general perturbaton expresson for the modal gan based on a vectoral eld soluton for the unperturbed wavegude wth arbtrary two-dmensonal cross-secton. Unfortunately, ths expresson, smpl ed to the case of a slab wavegude, cannot be recasted drectly nto a con nement factor formalsm. Several approxmate expressons for the TM modal gan, each of them based on a d erent con nement factor, wll be derved and explored n ths paper. It wll be shown that t s possble to calculate accurately the TM modal gan usng an expresson whch approxmates Vassallo's formula and whch s formally dentcal to the well-establshed TE formula and thereby retans ts practcal advantages. Fnally, the theory of the analytc approach wll be outlned and ts accuracy wll be proven. Ths approach has the same exblty n terms of wavegude desgn as the perturbaton formulaton. 2. TE versus TM modal gan Startng from the TE and TM wave equatons for the domnant eld components e y x and h y x of the one-dmensonal slab wavegude wth complex refractve ndex pro le n x, and wth z the propagaton drecton and k 0 the wavenumber n vacuum d 2 e y x 2 k0 2 n2 x b 2 e y x ˆ0 n 2 x d 1 1 dh y x n 2 k0 2 x n2 x b 2 h y x ˆ0 t can be shown, usng a varatonal approach that the magnary part of the square of the e ectve ndex sats es [9] Im n 2 x e y x 2 Im n 2 eff ˆ 2a e y x 2 for the TE case, and J. Haes et al. 264

3 TE and TM modal gan Im n 2 eff ˆ 0 Im n 2 x h y x 2 Im B h y x h y x dh y x dln n 2 1 x C h y x 2 A k0 2 for the TM case. It s seen that Im n 2 eff s gven by two d erent equatons. The rst term n (2b) s the drect analogue of (2a) and the second term occurs because the TM wave equaton descrbes a magnetc eld n a purely delectrc materal: the dervatve dh y x / s, unlke de y x /, dscontnuous at a refractve ndex step. In typcal III±V semconductor wavegudes, the domnant TE and TM eld pro les are very smlar. The d erence between TE and TM modal gan values s therefore domnated by the second term n (2b). As we wll see further on, ths d erence can be as large as a factor of four. Equatons 2 can be used n two ways. In the rst nstance they may serve as a check on the accuracy of a complex mode solver. Gven the calculated mode e ectve ndex, the eld pro les can be constructed and Expressons 2a and 2b should return (n an mplct way) to the e ectve ndex tself. A more practcal applcaton can be found when nterpretng Equatons 2 n a perturbatve way. esettng the magnary part of the refractve ndex pro le to zero, calculatng the real modes of the real ndex structure and substtutng the obtaned mode pro les n (2a) and (2b), whle takng nto account the magnary ndex pro le, gves a rst-order estmate of the magnary part of the e ectve ndex. However, Equaton 2b s n practce not very useful. Typcal nput data for laser modellng consst of the con nement factors of the domnant eld component n the d erent wavegude layers. It s mmedately clear that (2a) as well as the rst term of (2b) lend themselves to a con nement factor formulaton. 3. Con nement factor formulaton of TE and TM modal gan Consder a slab wavegude de ned by a real valued refractve ndex pro le n 0 (x) and suppose that a modal soluton (e 0 (x), h 0 (x)) s known. The wavegude s now perturbed by a pure magnary ndex contrast dn 00 x. Applyng standard perturbaton theory on Maxwell's two curl equatons t s shown n [10] that the rst-order correcton dn e on the effectve ndex of the mode s gven by r e 0 dn eff ˆ l 0 crosssecton crosssecton n 0 x dn 00 x je 0 x j 2 dy e 0 x h 0 x u z dy ˆ dn00 eff and s therefore purely magnary. If we lmt the dscusson to pecewse constant refractve ndex pro les wth layer ndces n 0,, the general perturbaton result (3) can be smpl ed n the TE case to gmod TE ˆ 1 X N n TE n 0; C e y g mat; 4 eff;0 ˆ1 where we ntroduced the modal gan g mod =2k 0 Im(n e )=2k 0 dn 00 eff and the materal gan of the th layer g mat; ˆ 2k 0 Im(n )=2k 0 dn 00 and N, the number of layers. C e y s the con nement factor of the domnant TE eld component n layer of the unperturbed structure 2b 3 265

4 It follows mmedately that C e y ˆ e y;0 x 2 layer X N ˆ1 e y;0 x 2 5 C e y ˆ 1 6 Gven the de nton of the real part of the longtudnal component of the Poyntng vector, Sz TE x ˆe S x u z ˆ1 2 e e 0 x h 0 x u z ˆ b0 2xl 0 e y;0 x t s seen that the TE modal gan can also be expressed by means of the power ux, as s frequently done (see for example [6]). The result (4) s wdely known and appled n practce, both for TE and TM polarzaton. One generally de nes a con nement factor based on the domnant h y eld component n the latter case. The fact that ths leads to erroneous results can be understood from (2b), as was explaned n the prevous secton. To express the TM modal gan the general result (3) s formulated explctly for a TM mode of a layered slab gmod TM ˆ X N ntm eff;0 ˆ1 layer n 0; e x;0 x 2 e z;0 x n 2 0 x e2 x;0 x g mat; 8 By analogy wth Equaton 4, the TM `con nement factor' should be de ned as e x;0 x 2 e z;0 x 2 2 e tot;0 x 2 C TM ˆ n TM layer 2 layer eff;0 n 2 0 x e2 x;0 x ˆ n TM eff;0 9 n 2 0 x e2 x;0 x Ths expresson, although exact wthn the perturbaton approxmaton, has two drawbacks. Frst, (9) does not descrbe a well-de ned physcal quantty wth a clear nterpretaton of eld con nement or power ux con nement, as was the case for TE. We wll now derve two approxmate expressons from (9) wth a physcal meanng and whch are true con nement factors n the sense of (6) [11]. As we are only nterested n guded modes, k x k 0. Hence 2 n 2 0 x ˆ kx 2 x 2 10 polarzaton. Second, the property (6) s not ful lled by C TM ntm eff;0 k 2 0 n TM eff;0 whch corresponds physcally to the replacement of the actual refractve ndex pro le by the average ndex as t s seen by the mode pro le. Furthermore, t follows from the TM equatons that the nequalty 266 J. Haes et al.

5 TE and TM modal gan e 2 z;0 x e 2 x;0 x ˆ dh y;0 x = 2 b 2 h 2 y;0 x O k2 x b holds for (well-) guded modes. Addng an extra term e z;0 x 2 n the denomnator leads to the con nement factor C TM;e tot e x;0 x 2 e z;0 x 2 C TM layer e x;0 x 2 ˆ C TM;e tot e z;0 x 2 12 whch approxmates very well wth the value of C TM. As we only consder unperturbed problems wth pure real ndex pro le, e x,0 (x) s also real valued and hence e 2 x;0 x ˆ je x;0 x j 2. Alternatvely, we can also neglect the e z;0 x 2 term n the nomnator to obtan e x;0 x 2 C TM layer e x;0 x 2 ˆ C TM;e x the con nement factor of the e x,0 (x) eld component. Ths approxmaton gets less accurate when the rato kx 2=b2 ncreases,.e. for the hgher-order guded modes of a multmode wavegude. To summarze, we now have three perturbaton formulae to calculate the TM modal gan gmod TM ˆ 1 X N n TM eff;0 ˆ1 n 0; C TM g mat; 13 1 X N n TM n 0; C TM;e tot g mat; 14 eff;0 ˆ1 1 X N n TM eff;0 ˆ1 n 0; C TM;e x g mat; The con nement factors of the last two equatons have a clear physcal meanng. The quantty C TM;e tot expresses the con nement of the total electrc eld leadng to the same formula for both TE and TM modal gan. In the case of very strong gudance, where the modal eld s completely con ned nto the core layer of the wavegude characterzed by a complex refractve ndex, (14) predcts that the hgher-order modes (wth lower n e ) experence a hgher modal gan. Ths result mght seem rather surprsng at rst glance, but can be ntutvely understood by consderng the ray pcture of modal propagaton. Hgher-order modes have a more skewed angle wth respect to the wavegude axs and therefore propagate a larger dstance n the core layer. They see more gan per unt dstance along the wavegude axs than a paraxal propagatng mode. The quantty C TM;e x s the con nement of the, for TM modes, domnant electrc eld component. Ths con nement factor has the advantage that t follows mmedately from the h y -con nement factor C TM;h y as 267

6 C TM;e x ˆ P N jˆ1. C TM;h y n 4 0; C TM;h y j. 15 n 4 0; j For wavegudes wth small ndex contrast t s clear that C TM;e x C TM;h y. In homogeneous space ths holds even exactly. The d erence between TE and TM modes vanshes n ths case, so that the e ectve ndex almost equals all of the refractve ndces. Hence, the rato between n 0, and n e,0 can be neglected n both Equatons 4 and 14, leadng to the result derved by [1] (only TE). It also follows that under the condtons of extreme gudance (e ectve ndex approaches the core ndex and the con nement n the claddng layers can be neglected) or when the mode evolves towards cuto (e ectve ndex approaches the claddng ndex and the con nement n the core can be neglected), that C TM;e x C TM;h y. Under these crcumstances, the modal gan calculated by C TM;h y wll gve an accurate result. Ths wll also be the case for the con nement of the power ux C TM;S z. 4. Analytcty of the dsperson relaton In the prevous secton we focused on a perturbaton approach to calculate the n uence of a small magnary perturbaton of the refractve ndex pro le of a slab wavegude on the modal e ectve ndex. In ths secton we wll explore an alternatve soluton. It s known that the dsperson relaton for the e ectve ndex as a functon of the wavegude geometry and ndex pro le s an analytc functon [8, 12] n those parts of the complex plane where there are no branch ponts. Suppose that layer wth ndex n 0, s perturbed by an magnary ndex dn 00. The mod ed e ectve ndex, due to the perturbaton, s gven by the Taylor expanson n eff n ˆn eff n 0; dn 00 ˆn eff n 0; dn 1 2 dn002 nˆn 2 n 2 O dn 3 nˆn 0; The dsperson relaton beng an analytc functon, the Cauchy±emann condtons e n e n Im n Im n e n Im n n e n are ful lled, expressng that the dervatve n the complex plane of the dsperson relaton n one refractve ndex pont s ndependent of the drecton. Practcally, ths means that the n uence of a small magnary perturbaton can be calculated by addng a real perturbaton to the ndex pro le, as s shown by the rst equalty n (17). It follows also from (16) that the rst-order correcton on the e ectve ndex s purely magnary (cf. (3)). Ths means that the trajectory of the effectve ndex n the complex n -plane wll move away from the real axs n a drecton parallel to the magnary axs. The dfference between TE and TM polarzaton s accounted for by the dependence of the dsperson equaton on the refractve ndex pro le. The thrd term n Equaton 16 suggests that the theory can be extended to hgher-order correctons. To evaluate ths term t would be preferable to use agan the analytc property (17). It s ndeed true that the eff =@n s analytc. Knowledge eff =@n n the drect vcnty of the nomnal refractve ndex pont n 0, makes t possble to evaluate the 268 J. Haes et al

7 TE and TM modal gan second-order correcton. Practcally, the e ectve ndex n three real ndex ponts around n 0, de nes a parabola. The coe cent of the quadratc term determnes the second-order correcton, whch s found to be real, see Equaton 16. In practcal calculatons, the dervatve n (17) wll be numercally evaluated, whch may lead to a loss of accuracy. It s therefore not obvous at rst sght f ths method predcts the modal gan correctly. From numercal smulatons we found that f the normalzed propagaton constant B ˆ n 2 eff n2 cl;2 = n2 cl;1 n2 cl;2, wth n cl,1 > n cl,2, s known up to three sgn cant dgts, the magnary part of the e ectve ndex s calculated wthn an accuracy of a few per cent, compared wth the result of a complex mode solver. Fnally, we remark that the theory has been presented for the case where the refractve ndex vares n only one layer. Where d erent layers are perturbed one could apply the aforementoned theory separately to each layer and add up all the e ectve ndex correctons, or one could alternatvely parametrze the refractve ndex pro le as n ˆ n 0; qdn where ˆ 1; 2;...;N and take the Taylor expanson wth respect to the parameter q. Evaluatng the rst-order term gves the mod ed e ectve ndex n only one calculaton. 5. Numercal examples 5.1. Modal gan calculaton We consder three d erent wavegude structures and calculate both TE and TM modal gan usng the d erent perturbaton approaches outlned n Secton 3. The numercal results are compared wth the gan values predcted by a complex mode solver and by makng use of the analytcty. The wavegude structures are chosen as representatve for a broad class of potental structures and are depcted schematcally n Fg. 1. The numercal results are summarzed n Table I for TE and Table II for TM polarzaton. Table I shows that the modal gan values for wavegude A calculated by the complex mode solver and the TE perturbaton formula concde. The error of the analytc calculaton s very small. Turnng to TM polarzaton, Table II reveals that only the perturbaton result usng G TM reproduces the mode solver result. It s also seen that the perturbaton approaches usng C TM;e tot and C TM;e x approxmate very well the correct soluton. Modal gan values predcted by C TM;S z and C TM;h y are wrong. The error of the analytc approach s agan margnal. The same concluson apples for the strongly asymmetrc wavegude B. We also notce that TE and TM modal gan values dffer by a factor of two for wavegude A and even a factor of four for structure B. The last example s the symmetrc trmodal wavegude C. Accordng to Table I, the TE perturbaton theory s correct for all three modes. For the TM case only the perturbaton result G TM concdes wth the mode solver value. The true con nement factor C TM;e tot s able to predct the modal gan for all three guded modes wth good accuracy and the Fgure 1 Wavegude geometres. The left wavegude s symmetrc, the rght strongly asymmetrc. 269

8 J. Haes et al. TABLE I Comparson between the modal gan calculated by a complex slab solver, TE perturbaton theory and the analytc approach. The numbers n parentheses ndcate the relatve devaton from the exact mode solver result Wavegude TE modal gan, cm ±1 Mode solver C TE;ey Analytc approach A B (±0.08%) C ± mode C ± mode (±0.05%) C ± mode (±0.06%) expresson based on C TM;e x underestmates the modal gan of the second-order mode sgn cantly. Ths s expected, for the e z (x)- eld component ncreases for hgher-order modes. We further remark that for all modes TE and TM modal gan values are comparable and that C TM;S z and C TM;h y lead to wrong results. The analytc approach proves agan to be sold. In every example the TE modal gan s larger than the TM modal gan, whch leads to the suggeston that bulk semconductor laser dodes wth bulk actve layers lase preferentally n the TE regme, not only because of the d erent mrror characterstcs for TE and TM, but also because of the d erence n modal gan Varaton of TM modal gan as a functon of the core thckness We now calculate the evoluton of the modal gan of the symmetrc wavegude structure of Fg. 1 as a functon of the core thckness d, and focus on the comparson of the dfferent TABLE II Comparson between the modal gan calculated by a complex slab solver, TM perturbaton theory, the dfferent TM con nement factor expressons and the analytc approach. The numbers n parentheses ndcate the relatve devaton from the exact mode solver result Wavegude TM modal gan, cm ±1 Mode solver C TM C TM;etot C TM;ex C TM;Sz C TM;hy Analytc approach A (±1.1%) (0.5%) (+35%) (+76%) (±0.04%) B (±1.4%) (±1.8%) (+44%) (+98%) (±0.39%) C ± mode (±0.2%) (0.2%) (+0.8%) (+1.2%) C ± mode (±0.7%) (0.2%) (+2.8%) (+5%) C ± mode (±0.8%) (±4.6%) (+4.5%) (+13%) (±0.07%) 270

9 TE and TM modal gan Fgure 2 Varaton of TM modal gan as a functon of the core thckness of the wavegude. On the scale of the gure, the modal gan calculated by a complex slab solver, the correct perturbaton expresson and the analytc approach are ndstngushable. The three curves correspond (from left to rght) to the fundamental, the rst- and the second-order mode. The performance of the dfferent con nement factor expressons are also compared. TM perturbaton approaches. The results are summarzed n Fg. 2. One may argue that t would be preferable to plot the normalzed propagaton constant versus the normalzed frequency. However, both quanttes are n ths case complex numbers and n the normalzed propagaton constant the real and magnary parts of the effectve ndex are not separable. From Fg. 2 t s concluded that the curves calculated by C TM and C TM;e tot are hardly dstngushable on the scale of the plot. The accuracy of the modal gan values based on C TM;e x decreases progressvely for hgher-order modes. For the fundamental mode, the d erence wth the C TM;e tot result s, as expected, neglgble. The con nement factor approaches based on C TM;S z and C TM;h y overestmate the modal gan sgn cantly. Only under the condtons of cuto or extreme gudance s the correct value approxmated Varaton of the effectve ndex as a functon of the materal gan We nvestgate now the accuracy of the analytc approach. Therefore we consder a monomode wavegude, typcal for semconductor optcal ampl ers at 1.3 lm and vary the materal gan of the actve layer n a very broad nterval gong from 10 cm ±1 to 10 4 cm ±1. Ths extremely hgh (and physcally unrealstc) upper boundary wll enable us to explore the lmts of the algorthm. The modal gan wll be calculated usng the analytc approach, perturbaton theory and a complex mode solver. The n uence of the gan on the real part of the e ectve ndex wll be calculated usng (16) and a complex mode solver. The varaton of the e ectve ndex as a functon of the core ndex n s approxmated, n the vcnty of the nomnal core ndex 3.60, for both polarzatons by the parabolas n TM eff ˆ 0:287n 2 1:778n 5:908 n TM eff ˆ 0:0915n 2 0:496n 3:804 de ned by the core ndex ponts 3.59, 3.60 and Substtuton of (19) n the Taylor expanson (16) leads to the curves labelled `analytc' n Fg. 3. The modal gan values obtaned by the complex mode solver, the analytc approach and the perturbaton theory concde perfectly, both for TE and TM polarzaton. The modal gan vares lnearly. Consderng the real part of the e ectve ndex t s seen that the expanson (16) predcts accurate results. Addtonal smulatons have shown that the numercal results are nsenstve to the precse choce of the core ndces for the calculaton of the parabolas (19)

10 J. Haes et al. Fgure 3 Varaton of real and magnary (expressed through the modal gan) parts of the effectve ndex of the fundamental mode as a functon of the materal gan of the core. The wavegude s depcted n (a). On the scale of the gure, the dfferences between the results obtaned usng a complex mode solver, the correct perturbaton formula and the analytc approach dsappear. TE results can be found n (b) and TM results n (c). 6. Concluson It has been shown from a smple varatonal calculaton (see [9]) that TE and TM modal gan behave fundamentally d erently. Therefore, one has to be very cautous when generalzng TE formulae to the TM case, as s frequently done n the lterature for the case of the con nement factor formulaton, whch relates the materal gan of the wavegude layers to the modal gan of the wavegude mode. Startng from a general perturbaton formula, an accurate expresson, correct up to rst order, for both TE and TM modal gan were derved. Unfortunately, the TM modal gan s not gven by a true llng factor formulaton. By approxmatng the TM relaton we have shown that both TE and TM modal gan are gven by the same expresson g mod ˆ 1 n eff;0 X N whch s exact for TE modes and approxmate for TM modes. 272 ˆ1 n 0; C e tot t g mat; 20

11 TE and TM modal gan An alternatve way of calculatng the modal gan to rst order has also been nvestgated. Ths method reles on the fact that the dsperson equaton for the e ectve ndex s an analytc functon of the refractve ndex pro le. The dstncton between TE and TM polarzaton s accounted for n the form of the dsperson relaton. Usng ths approach the e ect of an magnary ndex perturbaton on the real part of the e ectve ndex can be calculated very accurately. The advantage of both methods s that n order to obtan the complex e ectve ndex only a real ndex modal calculaton has to be done and that the e ect of d erent materal gan or loss levels can be drectly estmated wthout any addtonal smulaton. The valdty of the TM con nement factor approxmaton, as well as that of the analytc approach, has been numercally ver ed. Acknowledgements J. Haes and. Baets wsh to acknowledge partal support from the ACE-2069 and 2039 projects (UFOS and ATMOS). J. Haes holds a doctoral fellowshp from the Flemsh Insttute for the promoton of scent c±technologcal research n ndustry (IWT) and B. Demeulenaere from the Natonal Fund for Scent c esearch of Belgum (NFWO). Partal support (T. D. Vsser) by the Technology Foundaton (STW) and (H. Blok) by the Schlumberger Fund for Scence, Technology and esearch s also acknowledged. D. F. G. Gallagher (Photon Desgn) and C. Vassallo (France Telecom CNET) are acknowledged for nterestng dscussons. eferences 1. J. BUUS, IEEE J. Quantum Electron. 18 (1982) T. D. VISSE, H. BLOK and D. LENSTA, IEEE J. Quantum Electron. 31 (1995) D. BOTEZ, IEEE J. Quantum Electron. 14 (1978) S. ASADA, IEEE J. Quantum Electron. 27 (1991) G. P. AGAWAL and N. K. DUTTA, Long-Wavelength Semconductor Lasers (Van Nostrand enhold Elec- trcal/computer Scence and Engneerng Seres, New York, 1986). 6. M. J. ADAMS, An Introducton to Optcal Wavegudes (John Wley, Chchester, 1981). 7. V. L. GUPTA and E. K. SHAMA, J. Opt. Soc. Am. A 9 (1992) C. VASSALLO, IEEE J. Lghtwave Technol. 8 (1990) T. D. VISSE, B. DEMEULENAEE, J. HAES, D. LENSTA,. BAETS and H. BLOK,, accepted for publcaton IEEE J. Lghtwave Technol. 14 (1996) A. YAIV and P. YEH, Optcal Waves n Crystals (John Wley, New York, 1984). 11. J. HAES, B. DEMEULENAEE, T. D. VISSE, D. LENSTA, H. BLOK and. BAETS, Proceedngs of the European Conference on Optcal Communcaton (1995) E. SMITH, S. N. HOUDE-WALTE and G. W. FOBES, IEEE J. Quantum Electron. 28 (1992) P. M. MOSE and H. FESHBACH, Methods of Theoretcal Physcs (McGraw-Hll, New York, 1953). 273