Guided to leaky mode transition in uniaxial optical slab waveguides

1592 JOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. 11, NO. 10, OCTOBER 1993 Guided-to-Leaky Mode Transition in Uniaxial Optical Slab Waveguides Lluis Torner, Member, ZEEE, Jaume Recolons, and Juan P. Torres, Student Member, ZEEE Abstract-The guided-to-leaky hybrid mode transition in slab optical waveguides made on uniaxial crystals such as LiNbOs or LiTaOs is analyzed. Two different guided-to-leaky transitions have been identified, namely the ordinary cutoff and the extraordinary cut08 which occur when considering negative and positive birefringent materials, respectively. Analytical but transcendental expressions have been obtained, yielding the critical optical axis orientation, relative to the waveguide axis, above which the totally guided hybrid modes become leaky. The results indicate that the value of the critical orientation strongly depends on the waveguide parameters. The possibility of tuning this critical orientation to a desired value through the waveguide parameters is discussed. I. INTRODUCTION NTEGRATED optical technology typically employs uniaxI ial dielectric crystalline materials, such as lithium niobate (LiNb03) and lithium tantalate (LiTaOs), to obtain low loss optical waveguides. Several waveguide fabrication techniques have been extensively developed in past years and, as a result, the involved technology has become mature [l]. Modulators, switches, polarization controllers, filters, and correlators, among other high-performance devices for signal processing, signal routing and sensing, have been demonstrated in integrated form [2], [3]. Most of these devices are based on the excellent electrooptic and acoustooptical properties of LiNb03 and LiTa03. Crystal anisotropy plays also a very important role in many devices, for instance to obtain TE-TM conversion or to optimize phase-matching conditions for second-harmonic generation and other nonlinear parameteric interactions, but it is the central point for the anisotropy-based cutoff devices [4]. The principle of operation of the anisotropy-based cutoff devices is the passive as well as electrooptically induced guided-to-radiated mode conversion. Light-intensity (cutoff or mode excitation) modulators [SI and passive TE-pass [6] or TM-pass [7] polarizers belong to this category of optical devices. With a few exceptions [8], [9], in most cases the guided-to-radiated mode conversion between the TE and the TM modes, which propagate in birefringent waveguides when the propagation direction is parallel or perpendicular to the crystal optical axis, is considered and exploited [lo]-[14]. However, with the exception of the former symmetric configurations, all the modes supported by a slab waveguide made Manuscript received May 5, 1992; revised November 2, 1992. This work was supported by the Comision Interministerial de Ciencia y Tecnologia of the Spanish Government under grants PT89-0178 and TIC92-0094-C02-02. The authors are with the Polytechnic University of Catalonia, Department of Signal Theory and Communications, 08080 Barcelona, Spain. IEEE Log Number 9209997. on birefringent materials are of the hybrid type, with the six field components. In addition to the well-known totally guided modes, such waveguides suppport the so-called leaky modes. These are leaky guided modes, in contrast to their leaky unguided counterparts which occur in the isotropic case [15]. They are always lossy and energy leakage outside the guiding region. Totally guided modes as well as leaky modes guided by uniaxial thin dielectric films have been extensively studied since the early days of integrated optics, so now the main features concerning guiding properties are well established [ 161-[20]. In particular, leakage losses have been experimentally measured [21] and theoretically calculated [22]-[24] for a Ti: LiNb03 waveguide, and extensive calculations on the dispersion properties of totally guided modes as a function of the optical axis orientation, relative to the waveguide axis, have been reported [25]. On the contrary, to the best of our knowledge, the critical orientation at which the totally guided modes become leaky has never been analyzed. Our aim here is to address this point. Our goal is to identify the conditions for the guided-to-leaky mode transition (GLMT) to occur, for the common LiNb03 and LiTa03-based waveguiding structures, and to investigate the dependence on the waveguide governing parameters of the critical optical axis orientation at which the GLMT takes place. Ultimately, we show that the GLMT exhibits a potential interest for switching applications. This paper is organized as follows. Section I1 contains the background of our approach. It is devoted to the introduction of the waveguide configuration we will deal with and to a brief revision of the different type of possible guided modes. In Section I11 the eigenvalue equation for hybrid guided modes is obtained using the characteristic-matrix approach, and in Section IV we discuss the properties of these modes that are relevant to the GLMT. In Section V we focus on the guided-to-leaky hybrid mode transition. Two different situations are identified, namely the ordinary cutoff and the extraordinary cut08 and the expressions giving the value of the critical optical axis orientation at which the GLMT takes place are obtained. Section VI concerns the possibility of tuning the value of this critical orientation through the waveguide parameters. Finally, in Section VI1 we emphasize our main results. 11. GENERAL MODAL PROPERTIES The optical waveguide we consider here is shown in Fig. 1. The cover is an isotropic material with refractive index nc. whereas the guiding layer and the substrate are uniaxial 0733-8724/93$03.00 0 1993 IEEE ~ __ TORNER et al.: GUIDE-TO-LEAKY MODE TRANSMISSION 1593 I isotropic nC I anisotropic nos, ne, Fig. 1. Schematic of the waveguide structure. materials. The substrate is homogeneous, whereas the guiding film can be either homogeneous or have an inhomogeneous refractive index profile. In the first case D is the film thickness and in the second one it stands for the characteristic depth of the refractive index profile. We consider a geometry in which the crystal optical axes (2) are placed in the waveguide plane forming an angle 0 with the waveguide axis, and an identical orientation in both substrate and film has been taken. The propagation direction is taken along the x-axis and a timeharmonic dependence has been assumed, so that the fields at any point have the form exp b(Px - ut)], with ,d being the propagation constant. In the coordinate system that coincides with the principal axes of the uniaxial crystals (X, Y, 2)) the relative dielectric tensor is diagonal and takes the form where no and ne are the ordinary and the extraordinary refractive indices of the material, respectively. For an arbitrary orientation of the optical axes, the dielectric tensor is obtained by application of the appropriate rotation operator to (1). For the orientation considered here one gets a symmetric tensor whose nonvanishing elements write E,, = n," sin2 e + n," cos2 8 eyy = n: cos2 e + n," sin2 e E,, =n: ezy = (nz - nz) sin e cos 8 (2) In general, a waveguide such as the former one can only support hybrid modes, with the six field components. In the limiting cases 0 = 0' (X, Y,-cut; 2-propagating) and 8 = 90' (X, Y-cut; Y, X-propagating), the allowed guided modes break down in the usual TE and TM sets. The ody nonvanishing component of the electric field for modes is Ey; therefore the behavior of such modes is governed by eyy. Thus, at 0 = 0') TE modes are affected only by the ordinary refractive index of the difterent media forming the wave, hence such waves are pure ordinmy modes. Likewise, at 8 = 90') one has eyy = n:, and thus the TE modes arepure extraordinary modes. These two types of modes are referred to as 0-like modes and e-like modes, respectively. Concerning TM modes, they have the other two components of the electric field. In the coordinate system used here the predominant component is E,; hence, the behavior of these waves is basidly governed by eZz. At 0 = 90°, one has E=, = E,, = n:; thus at this orientation the TM modes are 0-like modes. On the other hand, at 8 = Oo, E,, = n:? whereas E*, = nz, so that Th4 modes are affected by both refractive indices. Nevertheless, as ez2 = n:, the TM modes behave as predominantly ordinary modes. Such modes are referred to as (0) -like modes. For an intermediate value of 8, the hybrid guided modes are a superposition of the ordinary and the extraordinary waves propagating in an unbounded uniaxial dielectric medium. In the case of an homogeneous waveguide, the fields in the film are a superposition of two ordinary and two extraordinary plane waves [Z]. A particular combination of these waves corresponds to a perfectly guided mode when both ordinary and extraordinary waves are totally reflected at the interfaces between the guiding layer and the surrounding media. Then, outside the guiding region in fields are evanescent and the guided power propagates parallel to the waveguide axis. This is a totdiy guided hybrid mode. Because the ordinary and the extraordinary waves propagate with different phase velocities, the condition of total reflection at the boundaries cannot always be satisfied for both waves. This means that only one of them will achieve total internal reflection, whereas the other will be only partially reflected. This results in radiation losses. The actual nonstationary guided wave can be approximately described by a hybrid guided mode propagating with attenuation, thus having a complex propagation constant. This is a leaky guided hybrid mode. The relative weight of the ordinary and the extraordinary waves in the hybrid guided mode is dictated by the eigenvalue equation. The hybrid modes which main contribution corresponds to the ordinary or extraordinary waves are referred to as [o]-like modes or [e]-like modes, respectively. The precise nature of the [o]-like and [e]-like modes in terms of totally and partiallv reflected waves has a very strong influence on the loss coefficient of leaky modes [18]-[24], [26]. This nature will also play a crucial role on the GLMT. According to the former discussion, the critical optical axis orientation we are looking for follows from the eigenvalue equation at the cutoff point for the ordinary or extraordinary waves. Therefore, next we obtain the eigenvalue equation for guided waves, with the help of the characteristic-matrix approach. 111. CHARA~~ERISTIC-M~RIX FORMALISM The key of the formalism runs as follows. Since we are looking for the eigenvalue equation for guided solutions, the fields in the cover and in the substrate must be evanescent. So7 they would have the form 1594 JOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. 11, NO. 10, OCTOBER 1993 These fields must obey the wave equation. In a lossless uniaxial dielectric medium it writes v2E' + k&.]E = V(V . E) (5) with ko being the free-space wave number. In the substrate, taking into account (2), the substitution of (4) into the wave equation leads to two possible solutions for the decay constant ys that correspond to the ordinary and extraordinary waves. One gets, respectively (7) where N E p/ko is the effective index and (8) nosnes nes (6) = Jnas sin2 6 + nzs cos2 6 In the isotropic cover these solutions degenerate and one obtains (9) - [N2 - n2 1/2 ccl According to (6) and (7), in the substrate the evanescent fields are a superposition of the ordinary and extraordinary waves, so that (4) becomes 2.,(z) = 2os exp [koyos(~ - .)I + Ees exp [koYes(D - .)I (10) The existence of the solutiys (6) an! (7) implies that all the components of the fields E,, and E,, can be expressed in terms of two of them. We have chosen as independent variables the y and 2 component sof the electric field associated to the ordinary and extraordinary waves, respectively. Then, the electric field in the substrate can be written as where, in order to render the expressions in a compact form, a matrix notation has been adopted. The coefficients A appearing in the above expressions follow from the wave equation. One gets (12) (13) Aox = - tan8, Aey = ?Aox 4s Yo S N Yes Yos Yo s Aoz =j-Aox, A,, = jyN In the cover, the double solution (9) gives an expression analogous to (11), but for the TE and the TM field components. One has According to the characteristic-matrix formalism, the fields (1 1) and (14) are continued across the intermediate dielectric medium by means of a characteristic matrix containing the field solutions. Thus, the required boundary conditions on the waveguide dielectric interfaces are verified if the field solutions appearing in the characteristic matrix join with the evanescent fields given by (11) and (14). Following the Berreman approach [27], the characteristic matrix T of a dielectric film of thickness D is defined as Q(2 = D) = T*(z = 0) (15) with e( 2) being a four-dimensional array containing the tangential field components. As is customary, we have chosen an ordering criterion in such a way that the transpose of 9 writes 9' = (E,, Hy , E,, H,). The tangential components of the magnetic field in the substrate and in the cover appearing in Q following immediately from (11) and (14) by means of Maxwell equations. One arrives at Yes Aey gs(z) = L [T:: Tes ] jN jNAey Eo, exp [koYos(D - .)I Ees ~XP [kores(D - .)I Here q is the characteristic impedance of free-space and use has been made of the definitions Substituting (1 l), (14), (16), and (17) into (15), a homogeneous equation system for E,,, E,,, E,,, and E,, is obtained, and the condition for having a nontrivial solution is that the determinant of the coefficient matrix vanishes. This condition is the eigenvalue equation. After straightforward manipulation it can be expressed as Tos[N34 - Aey(N14 + YesN13)l - Aox[TesN34 + Aey(N24 + YesN23)] + [N24 + TesN14 - YesAeyNlP] + Yos[N23 + ~esN13 + AeyNl~] = 0 (19) where with .. Here Til, with i, 1 = 1 - 4, are the elements of the matrix T, and ri has been defined by the formal expression ri -1 + (1 +j~)(&z + &4) (22) with Sil being the Kronecker symbol. The characteristic matrix of a homogeneous uniaxial dielectric film can be analytically calculated by using the 4 x 4 TORNER et al.: GUIDE-TO-LEAKY MODE TRANSMISSION formalism. The derivation follows a general procedure derived by Vassell [28]. An alternative approach based om the wellknown Cayley-Hamilton theorem has been reported more recently by Wohler et al. [29]. When the optical axis lies on the waveguide plane the characteristic matrix is given by a simple expression, which can be found in [26] andz[2%]. In the case of inhomogeneous structures the characteristic matrix must be obtained with the help of approximate methods or numerical techniques. Here we have used the multilayer staircase technique in which the inhomogeneous is considered a iinite set of thin homogeneous films. Then, the characteristic matrix of the whole structure follows from the product of all the matrices associated to each layer, according to the ordering criterion prescribed in (15). IV. HYBRID GUIDED MODES In this section dicusss the properties of the hybrid guided modes that are relevant to the GW. We have considered media with positive and negative birefriqence, using in both cases the typical parameters of the LiNbO3 and LiTaOpbased waveguides operating at X = 632.8 nm. First, we ark going to deal With a material with Wgative birefringence, such as LiNb03. In Fig. 2(a) we have plotted the effective indices of the hybrid modes guided by an homogeneous waveguide with the followkg parameters 1 2.2946 n,f = 2.2108 n, = 1.0 G1: nos = = 2.2866 nee = 2.2028 D = 3.0pm This is a multimode waveguide, with two modes for each polarization. When 8 = Oo (131\2), G1 supports the "€30 and the TE1 modes, which are o-like modes, tugether $&h Be TMo and TM1 ones, which are (0)-like modes. Due to the small birefringence of GI, modes of equal order -gate with very similar effective indices. When 8 increases from 0" , these modes become hybrid and some intervals of values of 6' appear where the guided modes exhibit very close effective index values, to such an extent that they are almost degenerate. These intervals will be referred to as near-degenetacy ngtdas (NDRs). On the other hand, the first-order malm at t9 = O' become leaky over a critical 8-value. This is tbe @ded-toleaky mode transition angle (GLMTA) 81 for each m&. fie lowest order modes, TEo and TMo at 6 = Oo, rema guided for all values of 8. At 8 = WO, the four modes becolhe again TE-TM. In order to identify the hybrid modes, we are using the notation by Knoesen, Gaylord, and Moharam [SI. For instance, the mode which at 8 = Oo is the TEo and at 8 = 90' is the TMo, is denoted as: [TEo, TMo]. The behavior of the effective indeg of the allowed hybrid modes as a function of 8 is summarized in the diagram in Fig. 2@). §ymbols used in this figure, which is referred to as an [o]-[e] diagram, correspond to the above-introduced notation and tilt arrows indicate near-degeneracy regions. Let us now focus on the [TEI, TEo] and the [TMI, TEl] modes. In the neighborhood of 8 = 0' thew modes evolve with 8 as [e]-like and [o]-like modes, respedvculy. When .8 increases, after the common NDR the roles become inverted, hence the mode [TMl, TEl] is an [e]-like mode when it It I I 1595 2.30 1 1 2.28 5 : c 2-24 3 3 .$ 2.26 ?z 0 2.22 2.20 I I I 0 30 60 90 optical & orientation (~~TWUU) (4 L [el - I4 (b) Fig. 2. (a) Effective ipdex of the hybrid modes guided by G1 as a function of the optical primtation. Cootinusus liges ipdicotetotaUy guid@ modes and dashed lines correspond to leaky modes. @) Io]-[e] diagram for GI. undetgws the GLMI: Similarly, the [TEl, TEo] mode, which evtdves as an [e] -&e de in the neighborhood of 8 = 0' and as an [o]-& mode beyond the first NDR, exbibits a second NDR, now with the [TMo, TMl] mode. Bdh modes also invert their nature in the common NDR, so that the first me becomes again an [el-like mode. To sum up, both modes that undergo the GLMT are [e]-lie modes. Next, we are going to deal with a waveguide made on a po4tiYe bireftiqent material. In Fig. 3(a) we have plotted the effective indices of the hybrid modes guided by a homogeneous waveguide with various parameters nof = 2.1856 n,f = 2.190 n, = 1.0 which correspond to a LiTaOs-based sample. The [o]-[e] --- ' With G2 is shows in Fig. 3(bz The point is that in tbis case ,the GLMT takes ploce through the [.Ilike madas. F& 3 ~~~ to a tyaic;+r LiTaQ3-bssed waveguide, but if the values of the waveguide piqmrs are } G2: {nos = 2.1834 ne, = 2.1878 D = 6.0pm 1596 2.190 , 2.188 R F: $ 2.186 3 'c' 'c' U 0 2.184 1 2.182 ! I I 0 30 60 optical axis orientation (degrees) (a) Fig. 3. (a) Effective index of the hybrid modes guided by GZ as a function of the optical axis orientation. (b) [.]-[e] diagram for Gz. slightly modified, the ordering of the effective indices of the TE-TM modes with equal order that occurs at 6' = 0" may reverse in relation to the one exhibited by Gz. In this case, the [.]-[e] diagram becomes identical to Fig. 2(b), but with interchanged TE-TM and [.]-[e] roles. Finally, Fig. 4 corresponds to a typical proton-exchanged LiNb03 waveguide, with various parameters I n,f = 2.2466 n,f = 2.329 nc = 1.0 G3: {n,, = 2.2866 ne, = 2.2028 D = 1.5pm Due to the negative increment of the ordinary refractive index (nof < nos) that is obtained by the proton exchange technique, this waveguide does not allow guided solutions at 0 = 0" [30]. Likewise, at 6 = 90°, only TE guided waves (e-like modes) are allowed. G3 supports four modes. Both TE2 and TE3 modes propagate with N < nos; thus, any angular deviation renders them leaky modes. The TEo and the TE1 evolve as [e]-like modes when 6' decreases from 90°, and both of them become leaky. JOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. 11, NO. 10, OCTOBER 1993 2.34 2.30 2 % c 'c' $ 2.26 -4 .c1 0 K 3 2.22 2.18 / ' 0 0 0 .. 0 .. TE3 ------ - - ---_. -- I I 30 60 s 3 optical axis orientation (degrees) Fig. 4. Effective index of the hybrid modes guided by GY as a function of the optical axis orientation. The dotted line stands for the cutoff locus .v = fk9(0). The [o]-[e] diagrams must be interpreted with caution, since the notations of [o]-like and [e] -like mode and near-degeneracy region come from somewhat intuitive insights. Near cutoff, the identification of these characteristics on the 6' - N plot is not easy; hence, the corresponding diagrams must be interpreted accordingly. The main usefulness of the [.]-[e] diagrams comes from the fact that they summarize the properties of a given waveguide concerning off-axis propagation conditions, so that a great deal of the information can be immediately read off from them. V. GUIDED-TO-LEAKY MODE TRANSITION The GLMT takes place when one of the ordinary or extraordinary waves forming the hybrid mode reaches its cutoff for total internal reflection, whereas the other remains totally reflected. Thus, two different cases must be distinguished: the cutoff of the ordinary wave and the cutoff of the extraordinary wave. The first case will be referred to as the ordinary cutoff (0-cutoff) and the second as the extraordinary cutoff (e-cutoq. Next we obtain the expression giving the exact value of the GLMTA in both cases. An alternative approximate procedure based on coupled-mode insights is given in the Appendix. When the ordinary wave reaches its cutoff for total internal reflection at the film-substrate dielectric boundary, its transversal propagation constant at the substrate vanishes; hence, one has yos = 0. Thus, according to (6) we can write o-cutoff + N = no, (23) Consequently, the transversal propagation constant of the extraordinary wave in the substrate region II I TORNER et aL: GUIDE-TO-LEAKY MODE TRANSMISSION 1597 Conversely, since here i. f = 0, at the extraordinary cutoff one has T~~ = 0. According to (7), this condition yields e-cutoff * N = nes(Ol,e) (25) Hence 22 nes - nos nzs sin2 el,e + n:, cos2 el,e (26) 70s le-cutoff = nos . sin el,e Expressions (24) and (26) lead to the following general conclusion: The ordinary cutoff, taken as partial cutoff, can only occur when considering negative birefringent crystals. The equivalent statement applies for the extraordinary cutoff and positive birefrignent crystals. A. Ordinary Cutoff At the ordinary cutoff, the substitution of (23) into the expressions for the coefficients A defined in (12), (13) yields 1 70s 1 Yos A,, lo-cutoff - +oo (27) (28) Aey Aez lo-cutoff - 2 + These divergences allow us to notably simplify the expression of the eigenvalue equation (19), since only the predominant terms must be retained on it. One arrives simply at (N14 -k 7esN13)o-cutoff = 0 (29) When all media forming the waveguide are isotropic, = 0 implies ^leS = 0. In this case, taking into account also the properties of the characteristic matrix of an isotropic film, (29) becomes TM TE which is the cutoff condition for TE and TM modes [31]. B. Extraordinary Cutoff ing that At the extraordinary cutoff the coefficients A write assumC. Allowed Values of 81 Generally speaking, the GLMTA can take all possible values in the range [Oo ,MO]. Nevertheless, for typical waveguides, 81 is bounded by a milximum or minimum value, characteristic of each waveguide. The obtainment of this bounding value runs as foilows. Since we are looking for either totally and leaky guided solutions, power flow must be parallel to the film-substrate interface at least for one of the two kinds of waves, ordinary and extraordinary, forming the hybrid modes. For the ordinary waves this condition implies whereas for the extraordinary waves it writes with nef(e) being defined as in (8). At 6 = Oo, (36) reduces to (39, whereas at e = 90' one has nef,es(9Oo) = nef,es, and (36) becomes identical to (35) but for the extraordinary refractive indices. There upon, since for the waveguides such as GI, one has nof > nos > ne,, and the 0-cutoff occurs as N -3 no,, the maximum value that can be reached by comes fmn the equality n,f(et,,o) = no#. This condition yields The same expression holds for G3, although now stands for the minimum allowed value of On the other hand, when considering waveguides made on positive birefringent crystals such as G2, one has nef > ne, > nof > nos. Therefore, since the e-cutoff occurs when N = nes(Ol,e), the maximum value which can be reached by verifies ne,(Olm,e) = n,f. Hence, one gets In view of (37) and (38), we notice that 81, depends on the film and substrate refractive indexes, but it is independent of the cover refractive index, as well as on the ratio D/X. Also, el,,,, (ei,,,) does not depend on ne, (nef), whereas &n,e (&n,o) does. VI. DISCUSSION The behavior of the GLMTA as a function of the various waveguide parameters has been investigated by solving (29) and (34). The outcome for typical structures is shown in Figs. In Fig. 5 we have plotted dt as a function of the ratio D/X, for a GI-like waveguide. The curves in the plot correspond to the different hybrid modes guided by the structure for each value of D/X. As this ratio grows, not only the number of TE-TM-allowed guided modes at 8 = 0,90° increases, but the nature of the hybrid modes modifies as well, since leaky modes convert into totally guided ones. Both effects occur with very smaU variations of D/X; hence, very sharp 5-10. JOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. 11, NO. 10, OCTOBER 1993 I I D/h Fig. 5. Guided-to-leaky mode transition angle for the hybrid modes guided by GI, as a function of the A-scaled waveguide thickness. Shaded region corresponds to the guided-to-radiated mode transition. Here and in Fig. 6, the unlabelled curve corresponds to a mode which is cutoff at 0 = 0'. 90'. 50 3 i? 3 40U L 'c, U Y cl 30E '% - 20e 'c' 10Od 2 Fig. 6. Same as in Fig. 5, but for Gz. changes in 81 take place. Higher order TE-TM modes become allowed at given critical thicknesses, the cutoff thicknesses, whereas the leaky-to-guided mode transformation is a gradual but fast process as a function of DIX. The cutoff thicknesses of the TE-TM modes taking place at 0 = 0,90° are given by simple expressions [4]. In the neighborhood of these cutoff thicknesses, some roots of (29) yielding 191,~ > appear. These roots correspond to the a total cutoff and evolve very fast from O1m,o to 90' as a function of DIX. The same comments hold as well for the curves in Fig. 6, which corresponds to a 90 2 t P 2 70 'c' Y Y e E 'Z - 6 .: 50 30 2 4 D/ Fig. 7. Same as in Fig. 5, but for GB. Here shaded region indicates forbidden values. From left to right: TEo-TE4. at 6' = 90O. 01 II I I 2.290 2.292 2.294 2.296 2.: film ordinari refractive index Guided-to-leaky mode transition angle for the hybrid modes guided by GI. as a function of the film ordinary refractive index. Fig. 8. G2-like waveguide. Finally, Fig. 7 corresponds to a protonexchanged sample such as GJ. In this plot, 191,~ starts at 90", and decreases assymptotically toward Olm,o when DIX grows. In order to display the behavior of 01 as a function of the various refractive indices involved, we have chosen a representative example for each case. The obtained results have been plotted in Figs. 8-10. As above, the global pattern of the plots, including fast variations of 01 near 191 = 0' and 191 = 01,, and multiple values of 81 with a fixed value of n,f or n,f, comes from both the greater number of allowed modes TORNER et aL: GUIDETO-LEAKY MODE TRANSMISSION 1599 0' V I 2.lb 2.185 2.186 film ordinari refractive index Fig. 9. Same as in Fig. 8, but for G2. and the leaky-to-guided mode transformation, which take place as the film reeactive indexes increase. The dent plots that ma be emphasized is the large,stti#s,exhibited by 61 with very small increments in the film refractivb @xes. Typically, an ihcrement of about An N 0.001 yields A91 N lo, and the larger the value of 6im, the larger At+. Away fnnn the cutoff point of the TE-TM modes, 61 depends almost linearly with nof, in the case of GI and Gz, and with n,f, in the case of G3. The values of the film refractive indices can be reasonably controlled in the fabrication process, so that 81 can be approxima@ly tuned to a desired value within a wide range. Modification of the film refractive index by modifying the temperature of the guiding film, for instance by optical absorption processes, seems to be an intaresting possibility. Typically, the thermally induced bulk refractive index variations at room temperature amount to dn/dT w lo-* - 10-50C for LiNb03, LiTaOs, LiI03, or BaTi03 [32], [33], and as a rule dn,/dT is larger than dn,/dT. In this line, it must be noted that the variations of 61 caused by an increment of a film refractive index are practically counkManced by the same increase of the corresponding substrate refractive index. Therefore, in order to obtain a notable modiication on the value of 61, an asymmetrical response on the film and substrate refractive indices to a temperature change should be desirable. Photoabsorption heating caused by totally guided modes would verify this requirement, since most of the guided wave power travels into the film region. VII. CONCLUDING REUARKS We have investigated the guided-to-leaky hybrid mode transition in planar optical waveguides made on uniaxial crystals like LiNbO3 or LiTaO3. 'bo different guided-toleaky transitions have been identified, namely the ordinary cutoff and the extraordinary cutofi which respectively occur when considering negative and positive birefringent materials, and analytical, yet transcendental, expressions yielding the film extraordinari refractive index Pig. 10. Guided-tderLy mode transition angle for hybrid modw guided by G3, as a Bvlctiop of the extrPordiBary refrsotive index. From left to right: TE&, TE1, at 0 = WO. CaiW qtid axis orientation above which the totally guided hybrid modas became leaky, 81, have been &tab&. The dependem;e of 81 on tbe waveguide thickness and on tbe film and substrate reftactive indices has been investigated, and fast variations have been found. Concerning the cover refractive index, 61 is nearly insensitive to the particular value of this parameter, provided that a low-index medium is deposited over the guiding film, but large variations arise again when considering nearly synmetrid structures. Actually, in a result opposite that of the proton exchange technique, titanium indiffusion yields inhomogeneous waveguides. In order to take into BccouBt this fact, graded-index versions of G1 and Gz, with a gaussian profile in both the ordinary and extraordinary refractive indexes, have been analyzed. Numerfcs indicate that the graded-index nature of the guiding film renders $1 to different values than its stepindex caunterpiut. For instance, concerning the [TEl, TEo] hybrid mode guided by GI, w 15.1' for a step-index film, whereas 12.8" in the graded-index case. We have made no attempt to do extensive calculations for the graded-index waveguides, but in representative cases, no significant differences have been found in the behavior of 61 with the involved waveguide parameters, in relation to the step-index case. Potential new interesting applications of the guided-toleaky mode transition come from the switching properties associated with the large variations of 6, caused by changes in the refractive indices involved. In suitable conditims, leaky modes exhibit large losses [18]-124], [%I, [30]; thus, sharp switching characteristics will arise. Success depends on the possibility of tuning the value of this critical angIe through the waveguide parameters, either by controlling the fahication procedure or by means of dynamical processes. Thermally induced refractive index changes caused by photoabsorption seem to be an exciting possibility to be investigated [34]. 1600 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 11, NO. 10, OCTOBER 1993 APPENDIX In this appendix we briefly outline an approximate procedure, based on the coupled-mode approach, to obtain the GLMTA. Key points run as follows (see, e.g., [SI). Since eZy << eyy, E,=, the hybrid modes guided by G1-G3 can be approximately regarded as weakly perturbed pure TE and TM modes. TE waves would render to [e]-like modes, whereas TM waves would correspond to the [o]-like ones. Accordingly, the eigenvalue equation (19) can be approximated by the well-known expression where the integer m stands for the mode operator, 61,2;~~ = I,&;TM = n;/nz, and 62;~~ = n;/n;. According to (2), the appropriate substrate and film refractive indexes write n2 = n: cos2 0 + n; sin2 8 for TE modes, and n2 = n: in the TM case. Finally, the ordinary cutoff takes place at NTE = no,, whereas the extraordinary one comes from the condition NTM = nes(8). The outcome expressions render once again transcendental, but easier, equations than (29) and (34), with their numerical roots exhibiting an excellent agreement with the exact solutions. The best fit occurs for G3-like samples and also for single-mode GI, G2-like waveguides. REFERENCES [l] R. R. A. Syms, “Advances in channel waveguide lithium niobate integrated optics,” Opt. Quantum Electron., vol. 20, pp. 189-213, 1988. [2] L. ThylBn, “Integrated optics in LiNbOs: Recent developments in devices for telecommunications,” J. Lightwave Technol., vol. LT-6, pp. 8474361, 1988. [3] J. v Ctyroky, R. Goring, J. Janta, W. Karthe, A. Rasch, M. Rotttschalk, and J. Schrofel, “Integrated Electrooptic Modulators and Switches in LiNbOs.” Kybernetika, vol. 26, pp. 171-190, 1990. [4] T. K. Gaylord and A. Knoesen, “Passive integrated optical anisotropybased devices,”J. Mod. Opt., vol. 35, pp. 925-946, 1988. [5] A. Neyer and W. Sohler, “High-speed cutoff modulator using Ti-diffused LiNbOs channel waveguide,” Appl. Phys. Lett., vol. 35, pp. 256258, 1979. [6] T. Findakly, B. Chen, and D. Booher, “Single-mode integrated-optical polarizers in LiNb03 and glass waveguides,” Opt. Left., vol. 8, pp. 641-643, 1983. [7] J. J. Veselka and G. A. Bogert, “Low-loss TM-pass polarizer fabricated by proton exchange for 2-cut Ti: LiNbOa waveguides,” Electron. Lett., [8] S. Yamamoto and Y. Okamura, “Guided-radiation mode interaction in off-axis propagation in anisotropic optical waveguides with application to direct-intensity modulators,” J. Appl. Phys., vol. 50, pp. 2555-2564, 1979. [9] S. T. Kirsch et al.,“Semileaky thin-film optical isolator,” J. Appl. Phys., vol. 52, pp. 319G3199, 1981. [lo] D. Marcuse, “Electrooptic coupling between TE and TM modes in anisotropic slabs,” IEEE J. Quantum Electron., vol. QE-11, pp. 1111 Y.-K. Lee and S. Wang, “Electrooptic guided-to-unguided mode converter,” IEEE J. Quantum Electron., vol. QE-12, pp. 273-281, 1976, 1121 Y. Okamura, K. Kitatani, and S. Yamamoto, “Electrooptic leaky anisotropic waveguides using nematic liquid crystal overlayers,” J. Lightwave Technol., vol. LT-2, pp. 292-295, 1984. 1131 H. Onodera, I. Awai, M. Nakajima, and J. Ikenoue, “Light intensity modulation based on guided-to-radiation mode coupling in heterostructure waveguides,” Appl. Opt., vol. 23, pp. 118-123, 1984. vol. 23, pp. 37-38, 1987. 759-767, 1975. [14] H. Onodera and M. Nakajima, “High-efficiency light modulator using guided-to-radiation mode coupling in a graded-index waveguide,” Appl. Opt, vol. 25, pp. 2175-2183, 1986. 1151 T. Tamir and F. Y. Kou, “Varieties of leaky waves and their excitation along multilavered structures,” IEEE J. Quantum Electron., vol. QE-22, pp. 544-551,- 1986. [161 D. P. Gia Russo and J. H. Harris, “Wave proDagation in anisotropic - => . .I thin-film optical waveguides,” J. Opt. Soc. Am., vol. 63, pp. 13glb5, 1973. 1171 W. K. Bums and J. Warner, “Mode dispersion in uniaxial optical waveguides,” J. Opt. Soc. Am., vol. 64, pp. 441446, 1974. 1181 J. \v Ctyroky and M. Cada, “Guided and semileaky modes in anisotropic waveguides of the LiNbOs type,” Opt. Commun., vol. 27, pp. 353-357, 1978. 1191 K. Yamanouchi, T. Kamiya, and K. Shibayama, “New leaky surface waves in anisotropic metal-diffused optical waveguides,” IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. 298-304, 1978. 1201 D. Marcuse and I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media. Part 11: Slab with a coplanar optical axis,” IEEE J. Quantum Electron., vol. QE-15, pp. 92-101, 1979. 1211 S. K. Sheem, W. K. Bums, and A. F. Milton, “Leaky-mode propagation in Ti-diffused LiNbOrr and LiTaOx waveguides,” ODt. Lett.. vol. 3, I .. pp. 7678, 1978. 1221 W. K. Burns, S. K. Sheem, and A. F. Milton, “Approximate calculation L> of leaky-mode loss coefficient for Ti-diffused LiNbOs waveguides,” IEEE J. Quantum Electron., vol. QE-15, pp. 1282-1289, 1979. [23] J. \v Ctyroky and M. Cada, “Generalized WKB method for the analysis of light propagation in inhomogeneous anisotropic optical waveguides,” IEEE J. Quantum Electron., QE-17, pp. 1064-1070, 1981. 1241 M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguides,” J. Lighfwave Technol., vol. LT-3, pp. 773-778, 1985. [25] A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol., vol. LT-6, pp. 1083-1104, 1988. [26] L. Torner, F. Canal, and J. Hernandez-Marco, “Leaky modes in multilayer uniaxial optical waveguides,” Appl. Opt., vol. 29, pp. 2805-2814, 1990. (271 D. W. Berreman, “Optics in stratified and anisotropic media: 4 x 4matrix formalism,” J. Opt. Soc. Amer., vol. 62, pp. 502-510, 1972. 1281 M. 0. Vassell, “Structure of optical guided modes in planar multilayers of optically anisotropic materials,” J. Opt. Soc. Amer., vol. 64, pp. 166173, 1974. [29] H. Wohler, G. Haas, M. Fritsch, and D. A. Mlynski, “Faster 4 x 4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Amer. A., vol. 5, pp. 15541557, 1988. 1301 J. Ctyrokjr, “Light propagation in proton-exchanged LiNbOs waveguides,”J. Opt. Commun., vol. 5, pp. 16-19, 1984. [31] L. Torner, F. Canal, and J. Hernandez-Marco, “Cutoff behaviour of graded-index slab waveguides,” Opt. Quantum Electron., vol. 21, pp. 451462, 1989. 1321 R. C. Miller and A. Savage, “Temperature dependence of the optical properties of ferroelectric LiNbOs and LiTaO3 ,” Appl. Phys. Lett., [33] D. W. Rush, B. M. Dugan, and G. L. Burdge, “Temperature-dependent index-of-refraction changes in BaTiOs .” Opt. Lett., vol. 16, pp. 1295-1297, 1991. [34] I. C. Khoo and P. Zhou, “Dynamics of switching total internal reflection to transmission in dielectric-cladded nonlinear film,” J. Opt. Soc. Amer. B, vol. 6, pp. 884888, 1989. vol. 9, pp. 169-171, 1966. Lluis Torner, photograph and biography not available at the time of publication. Jaume Recolons, photograph and biography not available at the time of publication. Juan P. Torres, photograph and biography not available at the time of publication.