characteristics of hybrid modes in proton-exchanged lithium niobate waveguides

Characteristics of hybrid modes inproton-exchanged lithium niobate waveguides

Giovanni Tartarini, Paolo Bassi, Pascal Baldi, Marc P. De Micheli,and Daniel B. Ostrowsky

A rigorous numerical model, verified by experimental results, gives an explanation of the particularelectromagnetic behaviors observed in x-cut proton-exchanged lithium niobate waveguides. Thisapproach, which allows an exact calculation of the weights of the coupled ordinary and extraordinarywaves that make up the hybridmodes, provides deeper insight into the study of the strains induced by theproton-exchange process in the waveguide itself, showing that the optical axis of the exchanged layer isnot parallel to the waveguide plane.Key words: Hybrid modes, proton-exchanged waveguides, anisotropic waveguides.

1. Introduction

Proton-exchanged lithium niobate 1PE:LiNbO32 wave-guides1,2 have been successfully used for many inte-grated optical applications. Among their interestingcharacteristics, the possibility of achieving high Dnewas mentioned, but none of the practical devicesrealized yet has used this property. Neverthelessthis property would allow the realization of sharpbends, which are of interest for increasing the densityof integration. It would also, in combination withother waveguide-fabrication techniques, allow one torealize highly asymmetricMach–Zehnder interferom-eters, which present interesting spectral propertiesand can be used as delay lines in coherent systems.Finally, the high-Dne waveguides are associated witha reduced electro-optic and nonlinear coefficient andcould be used in order to produce the periodic modula-tion of the nonlinear coefficient that is necessary toachieve quasi-phase-matched nonlinear devices.But these waveguides also present important stressesand negative Dno, which introduces a complex behav-ior and requires careful electromagnetic modeling inorder to explore or foresee the structure behavior.In a recent paper3 a completely general numerical

G. Tartarini and P. Bassi are with the Dipartimento di ElettronicaInformatica e Sistemistica, Universita di Bologna, Italy. P. Baldi,M. P. De Micheli, and D. B. Ostrowsky are with the Laboratorie dePhysique de la Matiere Condensee, Universite de Nice SophiaAntipolis, France.Received 8 June 1994; revised manuscript received 28 November

1994.0003-6935@95@183441-08$06.00@0.

r 1995 Optical Society of America.

model for the study of propagation in anisotropicplanar waveguides with arbitrary distributions of theordinary and the extraordinary refractive indices andarbitrary axis orientations was presented. The im-portance of a theoretical model in connection withexperimental measurement results is evident. Infact, fitting between experimental and theoreticalvalues of the effective refractive index of the modes1neff2 can be used to study the effect of the experimen-tal conditions on the waveguide characteristics, pro-viding, for example, estimates of ordinary and extraor-dinary steps 1Dno, Dne2 that guide layer width and tiltangle between optic axes in the guiding layer and thesubstrate. Modes of these guides generally werefound to be hybrid, resulting from phase-matchedpropagation of an ordinary and an extraordinarywave. The model was validated, and its capabilitieswere successfully exploited in Refs. 4 and 5 to inter-pret results obtained for real proton-exchange 1PE2waveguides fabricated in different experimental con-ditions. In the framework of this research, we ob-served possible particular and sometimes also unex-pected behaviors of these modes. For example, forcertain directions of propagation, the losses of somehybrid modes are shown to be higher for modes oflower mode number. So we investigated more accu-rately the electromagnetic properties of these wave-guides, finding evidence on the role of the ordinaryand the extraordinary contributions of the hybridmode and gaining some interesting new conclusionson the modifications of the crystal structure after thediffusion process.The modes of an anisotropic waveguide have been

generally classified as quasi-ordinary 1qO2 and quasi-

20 June 1995 @ Vol. 34, No. 18 @ APPLIED OPTICS 3441

extraordinary 1qE2 as, when planar waveguides areconsidered and the propagation direction is changed,the real part b of the generally complex propagationconstant bc follows a qO behavior 1b almost constantwith varying propagation angle2 or a qE behavior 1bvarying for varying angles2. This feature also allowsone to deduce qualitatively the relative importance ofthe ordinary and the extraordinary contributions thatgive rise to the hybrid mode. But this approximatemodel does not allow any deeper analysis, for ex-ample, of the effective relative weights of the ordinaryand the extraordinary parts of the mode. The avail-ability of a rigorous approach allows a more detailedanalysis of the waveguide behavior, providing evi-dence that the unexpected phenomena can be relatedto the interplay of the ordinary and extraordinarywaves that couple to give rise to the hybrid modes.In the following we briefly recall the model charac-

teristics. Then, using the above-mentioned classifi-cation of the modes, we show that qOmodes with veryhigh losses may exist, though they are not expected inguides with negative ordinary steps. More gener-ally, the importance of the ordinary waves is evi-denced in the dispersion characteristics and also inthe electric-field shape of qE modes. Finally, weshow that an appropriate modeling of the strainsinduced in the exchanged layer by the PE processallows one to understand the effect of the coupledordinary waves on the losses of some qEmodes. Thiswill also explain the particular behaviors that couldbe observed experimentally, confirming the validity ofthe approach.

2. Method of Calculation

The model developed for the analysis is illustrated inRef. 3, and therefore only its general characteristicsare reported here. We consider a planar PE:LiNbO3waveguide. Both waveguide and substrate are con-sidered uniaxial. Using the coordinate systemsshown in Fig. 1, we express the direction of the opticaxis C with respect to the 1x, y, z2 system both in theguiding layer and in the substrate, through the polarand aximuthal angles q and w, respectively. The xaxis is orthogonal to the interface planes of the planarwaveguide. Different values of the longitudinal prop-agation direction on the y–z plane 1angle d2 correspondto different values of the propagation constant, whichis generally complex, and bc 5 k01neff 2 jleff2, where lefftakes losses into account.A plot on the y–z plane of the guide and the

substrate values of the ordinary and the extraordi-nary indices 1see Fig. 2 for a qualitative plot that isvalid for a PE waveguide2 can help in describing thepossible behaviors of the modes of anisotropic wave-guides.4 Modes exhibit guided behavior when bothordinary and extraordinary waves are guided, i.e.,when they are oscillating in the guide and evanescentin the cladding. This cannot happen in PE wave-guides because of the negative step of the ordinaryrefractive index. Modes exhibit semievanescent 1SE2

3442 APPLIED OPTICS @ Vol. 34, No. 18 @ 20 June 1995

behavior when neff 1d2 falls into region 1 in Fig. 2 andone of the two coupled waves 1the ordinary in thiscase2 is evanescent in the core, while semileaky 1SL2behavior takes place for a certain mode when neff 1d2falls into regions 2 or 3 in Fig. 2; one of the waves 1theordinary in this case2 is radiating in the substrate.The value of the propagation constant results from

the solution of the dispersion equation obtained whenthe continuity conditions of the tangential compo-nents of the vectors E and H at the interfacesbetween the adjacent homogeneous layers that formthe waveguide are imposed. Then the electric-fieldcomponents of the modes can be found straightfor-wardly.

3. Numerical Results and Discussion

To develop our considerations, we refer to an examplethat corresponds to a real waveguide.3 The LiNbO3crystal is x-cut, and the C axis is parallel to the y axisof Fig. 1. The substrate and the guiding layer of thePE waveguide are both assumed to be uniaxial, with

Fig. 1. Reference frames used throughout the paper.

Fig. 2. Projections on the 1y, z2 plane of the wave-vector surfacesof guiding film and substrate for a PE waveguide and regions ofclassification for the modes of the guide. Solid lines, film; dashedlines, substrate. See text for details.

nosub 5 2.285, nesub 5 2.205, Dno 5 20.025, Dne 50.108, and the width of the waveguide is 2 µm. Thepropagation characteristics of the waveguide havebeen calculated for different values of the propatationangle d, for q 5 0° in the substrate, q 5 15° in theguiding layer, and w 5 90° both in substrate andguiding layer. This set of values was proven toprovide an excellent agreement between the theoreti-cally calculated and the experimentally measuredvalues of the effective refractive index of the modes,as is shown in Fig. 3. In the dielectric tensor,obtained after rotation, the diagonal elements do notvary substantially, while off-diagonal elements do notvanish but are still small 1one order of magnitudesmaller than the diagonal ones, at least2 and are thecause of the hybrid nature of the modes.Figure 3 is a Cartesian plot of the first quadrant of

Fig. 1, so the behavior of the modes can be identifiedas SE or SL according to the region where neff falls.The choice of the value for q is not critical for thevalues of neff, but it is shown to be important below,when the losses leff are considered.As was said in the introduction, hybrid modes have

been classified as qE or qO according to the behaviorof the real parts b of their propagation constants 1or tothe behavior of neff2 as a function of the propagationangle d. Such a distinction can be made at firstglance with Fig. 3: the qE modes have a behavior ofneff 1d2 similar to the extraordinary curve of the guid-ing film 1resulting from the projection of the ellipsoidbelonging to the wave-vector surface2, while the qO

Fig. 3. Calculated 1L2 and measured 1*2 values of the effectiverefractive index neff versus the propagation direction d for a PEwaveguide. The two extreme curved lines 132 denote the limits forthe extraordinary index of film and substrate: for angles largerthan approximately 60°, the substrate index becomes larger thanthe film index. The straight lines 132 on the contrary, refer to thevalues of the substrate 1upper line2 and film 1lower line2 limits of theordinary index.

modes have an almost constant value of neff, similar tothe ordinary curve 1which is a straight horizontalline2. Note that different behaviors 1SE or SL2may beexhibited by the same mode if its dispersion curveneff 1d2 is such that for different ranges of values of thepropagation angle 1d2 the mode is contained in differ-ent regions among those evidenced in Fig. 2.Let us consider the first qE mode 1qE02, whose

dispersion curve neff 1d2 is the closest to the upper limitof the values in Fig. 3. When d is lower thanapproximately 45°, this mode belongs to region 1, andtherefore it exhibits a SE behavior, as the extraordi-nary wave is guided in the classical sense, while theordinary wave is evanescent in all the layers. Thesame behavior is exhibited by the qE1 mode for dlower than approximately 28° 1see Fig. 3 again2.Modes in region 1 have also been classified as guided,6in the sense that they do not radiate power in thesubstrate. Their components are the same as thosereported in Fig. 4 for the qE0 mode. Note that thetransverse x and h components can be related to theprojections of the extraordinary E field along theaxes. In fact, the ordinary wave, which is evanes-cent everywhere, practically does not contribute.For qE modes of increasing order 1qE2 mode in the

example we are referring to; see Fig. 32, or, for a givenqE mode at increasing propatation angles 1d greaterthan approximately 45° for the qE0 mode, d greaterthan approximately 28° for the qE1 mode in theexample we are referring to2, the effective index of themode falls into region 2 1max5nesub, nofilm6 , neff ,min5nefilm, nosub62. We say that these modes exhibit aSL behavior of the first kind, as the effective index issuch that the ordinary wave of the mode is no longerevanescent everywhere, but it is evanescent in thefilm and uniform 1leaky2 in the substrate. When neff

Fig. 4. Components of the electric field for the qE0 mode for adirection of propagation d 5 5°. See text for details.


falls into region 3 1nesub , neff , nofilm2 1this happens ford greater than approximately 30° for the qE2 modeand for the modes represented by the subsequentdispersion curves in our example2, the ordinary waveis uniform also in the film. We say that these modesexhibit a SL behavior of the second kind, and theirmacroscopic feature is that the power losses leff 1d2 canbe lower than those of the SL modes of the first kind.In fact, in the latter case, no active power is guidedalong the propagation direction by the ordinary wave,while this is not the case for the former situation:both waves carry a fraction of power along z withinthe guide, thus preserving a little bit more of thepower propagating along z. This particular behaviorpredicted by theory has also been observed experimen-tally and is discussed further below.So far, we have considered only qE modes, namely,

modes in which the contribution of the extraordinarywave is much higher than the contribution of theordinary one. It was said in the introduction that asDno , 0, prevalently ordinary 1qO2 modes are notexpected to exist. This is not the case, as can be seenfrom observation of, for example, the almost constantline neff . 2.255 1dispersion curve of the qO0 mode2 inFig. 3, which denotes prevalently ordinary features.The shapes of the components of this solution withreference to the electric field are reported in Fig. 5 forthe propagation direction of 5°. The particular char-acteristics of this solution are evident: the mainelectric-field component is the x one with only onelobe. The h component, however, shows a three-lobed ripple superimposed on a similar one-lobe shape,denoting the contribution of an extraordinary wavewith 3rd-order spatial periodicity in the waveguide.The diverging tail in the substrate is a feature of leakymodes, which are not allowed by mathematics 1these

Fig. 5. Components of the electric field for the qO0 mode for adirection of propagation d 5 5°. See text for details.


modes should have infinite power2 but are allowed bypractical consideration: the modes are not excited atz 5 2` and so can never have infinite power.7Attenuation is found to be extremely high 1approxi-mately 1300 dB@cm2, confirming that the ordinarywave, which is radiating, has a prevailing weight withrespect to the extraordinary one. The existence ofthis mode can then be explained as follows: theordinary wave, considered alone, cannot exist as apropagating mode, but if it is coupled to a guidedextraordinary wave, it can exist 1in practice at leastfor a short while, as the high attenuation shows2.As the dispersion curve of this mode is an almosthorizontal line, as a consequence it seems to cross thecurve that represents the qE2 mode 1see Fig. 32. Thisbehavior is different from that presented by wave-guides with Dno . 0, where it can be shown that thecurves do not actually cross, but after having reacheda point of maximum proximity they diverge, to be-come well separated again.8 In this case the qOmode really ceases to exist for propagation directionsbetween approximately 34° and 44° 1see again Fig. 32.For angles larger than 44° the horizontal dispersioncurve refers again to the qO0mode, and the downwardsloping dispersion curve to the qE2 mode. The disap-pearance of the qO0 mode in the interval 334°, 44°4shows that the qOmodes in waveguides with negativeDno are critical. In fact, the qO modes disappearwhen their propagation characteristics are close tothose of a mode based on the extraordinary wave,which guarantees better guidance conditions. Weshow that this has consequences both for the qE2

mode field shape and, more important, for its attenua-tion characteristics. To do so we first follow thechanges in the field components of the qE2 mode forvarying propagation angles, in order to demonstratethe contribution of the ordinary wave to the modeitself. The qO0 mode, in fact, when it exists, exhibitspractically the same shapes of field components shownin Fig. 5, so no particular information can be gainedfrom its analysis.The one-lobe ordinary wave, whose contribution

has the prevailing weight in making up the qO0 mode,also contributes considerably to the qE2 mode, mainlyin the zone where the qO0 mode is absent. For d 5

26° 3see Fig. 61a24 the qE2 mode exhibits a SL behaviorof the first kind: in the guiding layer 122 µm , x , 02only the extraordinary wave has a uniform behavior1evidenced by the three lobes along both the x and theh directions2, while in the substrate 1x , 22 µm2 theordinary wave is radiating 1this is particularly evi-dent in the x component of the field2.For d 5 32° 3see Fig. 61b24 the qE2 mode exhibits a

SL behavior of the second kind: in the guiding layerwe can observe for the x component of the field acontribution from the one lobe ordinary wave thatadds to the three-lobed extraordinary one, while theqO0 mode still exists separately. For d 5 38° 3seeFig. 61c24 the qO0 mode does not exist, and the qE2mode presents a prevailing contribution from the

Fig. 6. Components of the electric field for the qE2 mode for a direction of propagation with 1a2 d 5 26°; 1b2 same as 1a2 but with d 5 32°; 1c2same as 1a2 but with d 5 38°; 1d2 same as 1a2 but with d 5 44°. See text for details.

one-lobe ordinary wave in its x component of theelectric field. Then, for d 5 44° 3see Fig. 61d24 qO0 andqE2 modes have crossed the respective dispersioncurves: now the effective refractive index of the qO0mode is higher than the refractive index of the qE2mode. The contribution of the one-lobe ordinarywave is again directed mainly to the qO0 mode, andconsequently its weight diminishes in the overallguided power of the qE2 mode.Something similar to what we have just observed

for the qE2 and the qO0 modes happens between theqE3 and the qO1 modes. The dispersion curve of theqO1 mode in this case disappears in a range of values

of the angle d, which includes d 5 0, and therefore wecan observe the almost horizontal dispersion curve ofthe qO1 mode only for a limited range of valuesaround d 5 28° 1Fig. 32. The behaviors of the electric-field components of the modes are as expected. TheqO1 mode, in the limited range of values of d where itshows up, presents a prevailing two-lobe ordinarycontribution in the guiding layer along x and aradiating behavior in the substrate. The qE3 mode,on the other hand, presents, in the range of values of dwhere the qO1 mode does not exist, a prevailingcontribution in its component along x of the two-lobeordinary wave, which is maximum for d 5 0 and


diminishes for increasing values of d. An example ofthe electric-field components of the qE3 mode for d 55° is reported in Fig. 7. So the qE3 mode is shown tobe 1until the value of d . 28°2 the combination of anextraordinary wave with 4th-order periodicity in thetransverse direction and an ordinary wave with 2nd-order transversal periodicity, the same as the qE2mode, which, in the range of approximately 334°, 44°4for the angle d, is shown to be the combination of anextraordinary wave of the 3rd-order with an ordinarywave of 1st-order transversal periodicity.A first conclusion that can be drawn is therefore

that ordinary and extraordinary waves follow theirtransverse resonance conditions independently andcouple to give rise to hybrid modes and that theirrelative weights in the hybrid mode are not constantbut vary according to the propagation direction.This result also has important effects on the attenu-

ation characteristics of qEmodes, the only modes thatcan practically propagate. Moreover, starting fromexperimental results, the model allows further under-standing of the structural changes induced in theLiNbO3 crystal by the PE process. Experimentalmeasurements9 show that in a certain range of valuesof d around d 5 0, the losses are lower for the qE3mode with respect to the qE2 mode. Our calculatedvalues model this behavior 1see Fig. 82: for angles dbelonging to a range of approximately 20°, whichincludes the value d 5 0, the losses of the qE3 modeare lower than the losses of the qE2 mode. This canbe ascribed to the fact that in such a range the qE2mode is SL of the first kind, while the qE3 mode is SLof the second kind, as has been explained above. Infact, for the qE3 mode the coupling of the two-lobeordinary wave increases the power that is guided inthe film, resulting in a lower value of the overall

Fig. 7. Components of the electric field for the qE3 mode for adirection of propagation d 5 5° and q 5 15° in the guidinglayer. See text for details.


losses of the mode itself. Note also that for the qE2mode a minimum of the losses is reached, because ofthe maximum coupling with the one-lobe ordinarywave. This happens for d . 38°, as has already beendescribed above 3see again Fig. 61c24.We remark on the crucial importance of having

assumed that q 5 15° in the guiding layer, whichmeans that the C axis is out of the guide surface.For comparison, Fig. 9 shows calculated values of

leff 1d2 obtained with the same parameters as in Fig. 8except the angle q, which is now also assumed to

Fig. 8. Behavior of the losses leff 1d2 for the qE2 and qE3modes for q

5 15° in the guiding layer. See text for details.

Fig. 9. Behavior of the losses leff 1d2 for the qE2 and qE3modes for q

5 0° in the guiding layer. See text for details.

equal zero in the guiding layer. In this case we showthat the losses of the qE3mode are always higher thanthose of the qE2 mode 1except near cutoff 2, and so thecalculated values do not model the above-cited mea-surements. Even allowing the C axis of the guidinglayer to rotate around the x axis 1q fi 90°2 does notchange this behavior, if the C axis is assumed to lie inthe waveguide plane 1q 5 0°2: the only effect of vary-ing w is to shift the two modes’ loss minima slightly.Therefore tomodel the experimentally observed behav-ior, it is necessary to assume that the C axis is out ofthe waveguide plane 1q fi 0°2 in the guiding layer.The reason for this can be understood when we

compare the field components computed for the qE2and qE3 modes with q 5 0°, w 5 90° in the guidinglayer 1i.e., in the case that does not model theexperimentally observed behavior2, with those com-puted for the same modes with q 5 15°, w 5 90° 1i.e.,in the case that models the experimentally observedbehavior2. This comparison can be done for anydirection of propagation d belonging to the angularrange in which the measured losses of the qE3 modeare lower than those of the qE2 mode. Choosing, forexample, d 5 5°, when q 5 0, w 5 90° we observe forboth the qE2 and the qE3 modes 1see Figs. 10 and Fig.11, respectively2 that the hybrid nature of the modesis not very important in this case: the contributionof the ordinary wave, which results in practice in avery slight radiating behavior of the x component inthe substrate, can hardly be appreciated for bothmodes. This explain why in this case 1see again Fig.82 the losses of both modes are rather small, and thelosses of the qE3 mode are higher than those of theqE2 mode for any propagation angle.


On the contrary, still for d 5 5°, taking q 5 15°, w 590°, we obtain the behavior already shown in Fig. 7for the field components of the qE3 mode and for thefield components of the qE2 mode behaviors, whichare practically the same as those shown in Fig. 61a2.We observe that in this case the hybrid nature of themodes is very important; i.e., a stronger coupling ofthe x-polarized two-lobed ordinary wave with thefour-lobed extraordinary wave takes place and canalso be appreciated in its consequences. In the qE2mode 3Fig. 61a24 we show that the contribution of theordinary wave results in amajor increase of the powerradiated in the substrate, because of the fact that theordinary wave is evanescent in the guiding layer.In the qE3 mode instead 1see Fig. 72, as has alreadybeen noted, the contribution from the strongly coupledx-polarized two-lobed ordinary wave increases notonly the power radiated in the substrate but also theguided power 1SL behavior of the second kind2, result-ing in a reduction of the losses of this mode withrespect to those of the qE2 mode.So the analysis shows that the coupling of the

ordinary and the extraordinary waves of the hybridmodes plays a crucial role in explaining experimen-tally observed phenomena, and a correct modeling ofthis behavior can be done only if the ordinary part ofthe mode is strongly coupled to the extraordinarypart. This happens only if the angle q may exhibit anonzero value in the guiding layer. This means thatin order to model correctly the electromagnetic behav-ior of LiNbO3 waveguides after the strains inducedby the PE process, it seems to be necessary to allowthe C axis of the exchanged layer not to lie in thewaveguide plane. We are currently analyzing thisparticular aspect, because it seems to play a basic role



in the complete characterization of real PE wave-guides, which is one of the aims of the main researchmentioned in the introduction.

4. Conclusions

In this paper we have explained and analyzed indetail possible mode shapes and loss characteristicswith respect to the propagation direction d that wehave observed for the SL modes of anisotropic PEwaveguides. These observations have been carriedout in the framework of more general research activ-ity, both theoretically and experimentally, that ap-plies a rigorous electromagnetic numerical method tothe modeling of PE LiNbO3 waveguides. We haveshown that despite the negative step of the ordinaryindex, qO modes can exist in these waveguides.However, they disappear when their dispersion curveapproaches the curve of a qE mode, and, as can beexpected, exhibit very high losses (approximately1300 dB/cm).Possible different spatial periodicities in the compo-

nents of the electric fields of ordinary and extraordi-nary coupled waves for the same hybrid mode havebeen demonstrated.In particular, the ordinary part seems to play an

important role in causing losses of modes with highermode number to be lower than losses of modes withlower mode number, and this is intimately related tothe fact that a correct modeling of the electromag-netic behavior of PE waveguides requires that thestrain induced in the guiding layer by the PE processmay cause the optic axis C of the crystal not to beparallel to the waveguide plane any more. We arecurrently engaged in deepening the analysis of thislast aspect.


Part of this work has been sponsored by the ItalianMinistry for the University and Scientific and Techno-logical Research. Acomplementary sponsorship pro-vided by the Italian National Research Council isgratefully acknowledged as well.

References1. J. L. Jackel, C. E. Rice, and J. J. Veselka, ‘‘Proton exchange for

high index waveguides in LiNbO3,’’ Appl. Phys. Lett. 41, 607–609 119822.

2. M. P. De Micheli, J. Botineau, P. Sibillot, and D. B. Ostrowsky,‘‘Fabrication and characterization of titanium induffused pro-ton exchanged 1TIPE2 waveguides in lithium niobate,’’ Opt.Commun. 42, 101–105 119822.

3. G. Tartarini, P. Bassi, S. Chen, M. P. De Micheli, and D. B.Ostrowsky, ‘‘Calculation of hybrid modes in uniaxial planaroptical wave guides: application to proton exchanged LiNbO3

waveguides,’’ Opt. Commun. 101, 424–431 119932.4. S. Chen, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, A.

Leycuras, G. Tartarini, and P. Bassi, ‘‘Loss mechanisms andhybrid modes in high dne PE planar waveguides,’’ Opt. Lett. 18,1314–1316 119932.

5. S. Chen, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, and A.Leycuras, G. Tartarini, and P. Bassi, ‘‘Hybrid modes in protonexchanged waveguides realized in LiNbO3, and their depen-dance on fabrication parameters,’’ IEEE J. Lightware Technol.5, 862–871 119942.

6. A. Knoesen, T. K. Gaylord, and M. G. Moharam, ‘‘Hybrid guidedmodes in uniaxial dielectric planar waveguides,’’ IEEE J.Lightwave Technol. 6, 1083–1104 119882.

7. A. W. Snyder and J. D. Love, Optical Waveguide Theory1Chapman and Hall, London, 19832.

8. L. Torner, F. Canal, and J. Hernandez-Marco, ‘‘Leaky modes inmultilayer uniaxial optical waveguides,’’ Appl. Opt. 29, 2805–2814 119882.

9. S. Chen, ‘‘Modes hybrides dans des guides realises par echangeprotonique sur LiNbO3,’’ M.S. thesis 1Universite de Nice, Nice,France, 19922.

characteristics of hybrid modes in proton-exchanged lithium niobate waveguides

Documents