THIN FILM EPITAXIAL OXIDE OPTICAL ...

THIN FILM EPITAXIAL OXIDE OPTICAL WAVEGUIDES D. K. FORK, F. ARMANI-LEPLINGARD, J. J. KINGSTON, and G. B. ANDERSON Xerox Palo Alto Research Center, Palo Alto, CA 94304, USA

ABSTRACT One of the most challenging applications of ferroelectric thin films is the formation of technologically practical optical waveguideing devices, particularly in the context of a dynamically changing environment where competing light sources and optical materials simultaneously undergo rapid improvement. In order to assess the prospects of this technology, a fundamental understanding of waveguide loss is being pieced together. This includes the relative contributions of surface scattering, and grain boundary scattering to optical losses. With computational models, it is possible to predict the surface losses from measured topographic data. This tool provides a method to probe the residual effects of grain boundaries, defects and impurities on optical losses. A comparative anatomy of various thin film structures and their loss characteristics will be provided in the context of these experiments. INTRODUCTION Electrooptic devices such as fast (>20 GHz) modulators are one application' of ferroelectric oxide thin film waveguides. A compact, blue laser source of a few milliWatts power capable of lasting thousands of hours is of great interest as applied to optical data storage and xerography, 2 and is the primary focus of our research. Ferroelectric oxide thin films offer several

advantages over bulk materials for optical waveguides, though no devices superior to bulk devices exist yet. One advantage is the larger refractive index difference between the ferroelectric layer and the cladding, which are dissimilar materials, such as lithium niobate and sapphire. Bulk devices rely on ion-exchange to produce an index change, which produces only a small index

difference. Thin films therefore permit higher intensity per unit power in the guide, and hence larger non-linear effects, and shorter interaction lengths. To be effective, the thin film devices must meet several challenges. Cost and performance in comparison to bulk devices are only two. One major challenge awaiting all frequency doubled blue sources is direct blue laser sources, particularly in light of the report of efficient gallium nitride light emitting diodes3 and the great research interest in gallium nitride lasers. Our research on thin film ferroelectric waveguides has concentrated on lithium niobate and lithium tantalate thin films on sapphire. An earlier publication detailed the external conversion efficiency of such waveguides as a function of film thickness. 4 To summarize these results, the optimum film thickness is about 400 nm, and the theoretical conversion efficiency of an ideal, lossless device is over 100 times larger than the present state-of-the art in bulk devices. There is a great challenge in exploiting this property however since for a frequency doubling device relying on copropagating modes, losses of 2 dB/cm reduce the conversion efficiency by about 50%. Another advantage of thin film optical heterostructures, is the ability to grow waveguides on diverse substrates, such as semiconductors. Such structures could have a packaging cost advantage over hybrid devices. Several groups5 6' ' 7'8 have worked on these structures. Where appropriate, particular loss considerations for these structures will be pointed out. In general, the growth considerations are far more demanding than for structures grown on a cladding substrate. The reward however is in principle greater since packaging costs of coupling together hybrid 189 Mat. Res. Soc. Symp. Proc. Vol. 392 ©1995 Materials Research Society

devices consisting of separate laser and frequency doubling devices are considerable, and would probably rule out their use in large volume laser disc or xerographic applications. Thin film waveguides of ferroelectric materials have been studied for over twenty years, however, these studies have failed to demonstrate reliably low loss waveguides with bulk-like optical properties. Furthermore, the advantages of thin films over bulk have not been reduced to practice. Understanding the cause of optical losses in detail is an important first step to solving the associated materials problems. WAVEGUIDE PROPAGATION LOSS An understanding of the origins of the optical losses is essential to optimizing waveguide performance. Separate loss mechanisms include absorption, leakage, internal scattering, surface scattering, and interface scattering.

Absorption Absorption may occur due to defects within the grains or at grain boundaries. Impurities and oxygen vacancies for example can cause absorption. For potassium niobate filns grown by liquid phase epitaxy 9, it was reported that blue colored layers could be rendered colorless by oxygen annealing or by adding 0.5 mol% ZrO 2 or HfO2 during growth. We have confirmed in our growths of lithium niobate thin films on sapphire that inadequate oxygenation, either accidental or intentional, also increases the propagation loss at 633 nm, presumably due to absorption. Since the absorption is proportional to the power fraction contained in the guide layer and the effective ray-optic path length, the absorption loss will increase with the guide thickness, decrease with wavelength, and decrease with the mode order. The variation is largest near cutoff because away from cutoff, the power fraction varies at an approximately equal and opposite rate as the path length. Near cutoff, the power approaches zero, and the path length increase

approaches

1ngud0isubstralc.

At visible wavelengths, material absorption usually increases with

wavelength. This is true for both oxides and polymers. Leakage Loss Cladded semiconductor substrates such as Si and do not support true guided modes, since the large refractive index of the substrate (n - 3.8 @ 633 nm) implies that a mode in the film will eventually leak into the substrate. This leakage is dependent on the thickness of the film and the cladding layer. Calculations of the required cladding layer thickness needed to limit the leakage loss to I dB/cm in a LiNbO 3 thin film clad with MgO on GaAs have been reported previously. 4 The results showed that for a wavelength of 1000 nm and the optimal film thickness of 400 nm, the cladding layer would need to be on the order of one micron or thicker. Growth of an epitaxial layer this thick with adequate smoothness presents a very difficult challenge. Scattering Loss Scattering, is in essence, the coupling of one mode to other guided modes and/or radiation modes. Variations in refractive index will induce scattering in the waveguide. This can occur either through material inhomogeneities or by grain misorientation in a thin filhn of birefringent material. In all types of films, inhomogeneities may result from amorphous inclusions or undesired phases such as the pyrochlore phases of lead titanate and potassium niobate, and LiNb 30 8 and Li3 NbO 4 in the case of lithium niobate. PbTiO 3 for example, may unavoidably contain inhomogeneities due to a mixed a/c orientation texture. Potassium niobate films have been reported to have either tetragonal1° or orthorhombic" crystal symmetry at room temperature, possibly due to a large shift in Curie temperature in the tetragonal films. High losses may result 190

from twinning in the orthorhombic films. Even highly aligned grain boundaries may pose a problem, due to strain or interfacial disorder. This work is among the first directed at separating "optically clean" grain boundary types from others. Scattering may occur both internally, or at surfaces and interfaces. These processes differ, for example internal scatter and absorption have the similar dependencies on film thickness, mode order and wavelength since they both depend on the amount of power in the guide layer, whereas the surface and interface scatter depend on the local mode intensity. The interface is typically, but not always, a polished substrate surface, with RMS roughness less than I nm. The guide layer surface, however, is typically a deposited film surface. Film morphology usually creates roughness far larger than that of a polished substrate. The scattering loss varies quadratically with the roughness. This stands a good chance of overshadowing the effect of there being a higher interface mode intensity in an asymmetric waveguide such as lithium niobate on sapphire. As reported earlier,4 for a given surface microstructure, surface and interface scattering losses peak at some intermediate thickness. This is due to the rise and fall of the mode intensity at the interfaces as the mode shifts from traveling mostly in the cladding to mostly in the guide layer as the thickness is varied. Lithium niobate films on sapphire or MgO which have optimal thickness for frequency doubling (;400 nm) are more sensitive to surface roughness than ion exchanged bulk devices in which modes penetrate several microns. The 400 nm thick films, however, are about twice as thick as the most surface sensitive thin film waveguides. Inhomogeneities within the bulk of the filn typically produce small but distributed variations of refractive index. Thickness variation, i.e. roughness and outgrowths in thin film waveguides, produces a large and localized variation in refractive index (for TE modes the surface scattering is proportional to the squared difference of squared indices). Thin filn structures which seek to exploit the large refractive index difference attainable via epitaxial growth will as a result be more susceptible to surface and interface scattering. Scattering due to thickness variation places very tight demands on the material processing. In the case of oxide films grown on lattice mismatched, dissimilar materials such as KNbO 3 on MgO or LiNbO 3 on MgO on GaAs, the microstructure may consist of small grains. The morphology of the grains will greatly impact the losses. What is required are nearly perfectly flat surfaces and interfaces. Model simulations below will place this assertion on a more quantitative footing. SURFACE LOSS CALCULATIONS Whereas the internal losses of a waveguide are difficult to assess, the roughness of a film or growth surface is easily measured by atomic force microscopy. This section highlights model calculations of surface losses based on measured film data. In a single mode guide, scattering losses consist primarily of power coupled into the continuum radiationmodes of the guide, and to a lesser extent to the back reflected guided mode. In multimode guides, mode conversion to other guided modes also contributes to the loss. Figure 1 shows a diagram of the allowed propagation constants (effective wavenumbers) of the guided and radiation modes of an asymmetric waveguide. Propagation constants range from -nik to nik where n, is the index of the film and k is the freespace wavenumber (21f!/2) of the light. The guided modes range between n2k and nlk where n 2 is the substrate index. The radiation modes (i.e. those modes which scatter light out of the guide) range from -n2k to n2k. The substrate modes loose light into the substrate only. To first order, for an inhomogeneity to scatter 191

light from one mode to another, it must have a spatial frequency equal to the difference in propagation constants of the modes. The Fourier transform of the inhomogeneity is useful therefore in predicting the scattering in the guide. One sees that scattering events which reverse the direction of a guided mode require a large scattering vector, which typically makes their contribution to the loss smaller than forward scattering. Reverse Reverse Guided Substrate Modes Modes

'0

Scattering Vector

Air Modes

. . . . . ..

q =130

Forward Forward Substrate Guided Modes Modes

1 Propagation Constant

3

Figure 1. Propagation constants of the various guided and radiation modes in an asymmetric slab waveguide carrying light of freespace wavevector k, n1 , n 2 and n 3 are the film, substrate and air refractive indices respectively. Surface roughness with spatial frequency q will scatter radiation

between a mode of propagation constant/50 and ,8. Our loss calculations employ the mode coupling formalism developed by Marcuse for asymmetric slab waveguides.12 The formalism predicts that the losses vary as the square of the RMS roughness. A very high premium therefore is placed on creating very smooth surfaces. Previously, estimation of the losses due to surface imperfections required making assumptions about the functional form of the surface roughness. 13 With atomic force measurements, such assumptions are no longer necessary. However, due to several factors including the quadratic dependence on roughness, the first order nature of the calculation, and the treatment of the boundary, accuracy in predicting the surface loss is limited to about a factor of two. For a multimode guide, the power fraction lost from the TE 0 mode to the n'/' higher order mode by surface scattering is given to first order by 2 AP k4162 (n - li1) A2 kAj(/ (2) 0L 0 /), where k is the free space wavenumber, / is the length of the guide, All is the power normalized amplitude of the uth mode at the top surface, and ý9is the spectral scattering power, given by

qo(,8 -, /)- fdx.-h(x)e

'•G•o)X

(3)

where h(x) is the surface topography of the waveguide. The scattering power, ý0,is computed from the A.FM data. One sees from (2) that a number of factors influence the loss. A large index change at the top surface will enhance losses, as will a large mode surface intensity. Since the surface intensity depends on the guide thickness (Figure 6), and wavelength (Figure 3), the losses do not vary strictly as i/,4 which is the familiar power law dependence for Rayleigh scattering of light from small particles. The expression in (3) varies in inverse proportion to root length, so the overall loss varies linearly with length. Consideration of the cumulative impact of losses over many infinitesimal lengths results in the exponential decrease in guided power that is observed in practice. 192

For most purposes, the continuum of radiation modes account for a large fraction of the surface losses. The expression for this loss component is,

k 4 12

tPrad

P

AI12V:

= 16 - (n2 -n2)2A,',n2k

(,6)20(/ d

' P8

,

(4)

0 -

which is very similar to (2) except that the loss must be integrated over the continuum of radiation modes. p is related to /3by the relation,

(5)

2

As indicated above, the band of radiation modes extends over the shaded part of Figure 1. The summation occurs over the pseudo-even and pseudo-odd continuum modes. 14 The same remarks about the index, mode amplitude and wavelength dependence for the guided losses (2) apply to the continuum losses LiTaO /MgO(l111)/AI 0 -c E (4). Slightly more complex A 2 L-a3 formulae apply to the TM 426 nm Thick ""modes.12 4.9 nm RMS surface roughness 1n0 a 235 nrn correlation length

•

Grain Size

0~

0

Figure 2 shows the scattering similar for 7. a filhn power in to in (3) Figure Shading

o0.5

03that

L)

09 0.0 0.00

0.02

0.04

0.06

0.08

0.10

0.12

Wavenumber q (nm-1) Figure 2. Fourier transform scattering power, qo,for an image like that in Fig. 7. The shaded band shows the spatial frequencies which allow first order scattering for modes at 633 nm.

Figure 2 corresponds to the band of allowed scattering vectors depicted in Figure 1. Scattering power outside this band can only contribute to the loss through higher order scattering processes. The rolloff of scattering power with frequency

is typical of all films, however, the bandwidth depends on the grainsize of the film. Scattering power for two films with differing morphology has been compared earlier.15 The correlation length of the roughness, ýc, which is a measure of the average grain size, is the first moment (q) of the scattering power, 2

(q)=Jq.qp(q).dq J(o(q).dq

(6)

Expression (6) gives an approximate measure of the spectral bandwidth of the scattering power due to the film roughness. When this bandwidth is on the order of the bandwidth of allowed propagation constants shown in Figure 1 (shaded region of Figure 2), the losses are maximized for a given roughness. For single mode or low mode films, this condition is met when the film thickness is on the order of the correlation length. Films which have surface variations of

higher or lower spatial frequency would generally have lower losses. Since polycrystalline thin films tend to develop a columnar microstructure, often with grain size on the order of the film thickness, the natural growth habit of most thin polycrystalline 193

50

LiTaO /0g0( 111)/Al 0I-c S".

" 40 30

Th 426 nm Thick 4.9 nm RMS surface roughness .... -TE 235 n• correlation length.

U)

2n 20 10 10O)0

900

800

700

600

0

40

Wavelength (nm)

films will tend to enhance the optical losses. Notice that the first order scattering band in Figure 2 contains a large fraction of the total roughness power spectrum! Uniform, single crystal films without grain boundaries thus have the greatest potential for acceptably low loss. An alternative is films with fine grained morphology, though the as yet poorly understood effect

of grain boundaries on internal loss would have to be minimal this strategy to work. for Figure 3. Calculated surface scattering loss versus wavelenoth for the data in Figure 2. The pointer indicates measured loss at 633 nm. Summing the terms in equations (2) and (4) for a surface micrograph, gives the total loss due to guided and radiation mode coupling. This can be done versus wavelength, and also versus thickness, if one is willing to accept the unphysical situation of unchanging roughness with changing thickness. Figures 3 and 4 display these results for the data in Figure 2. As was indicated above, the dependence of surface loss with wavelength is not expected to be 1/24 as is Rayleigh scattering. In the example in Figure 3, the behavior is closer to 1/2. The magnitude of the result is interesting as well. Above it was remarked that for frequency doubling applications, 2 dB/cm losses reduced performance by 50%. The losses at 1000 nm and 500 nm are both well above that limit. The RMS roughness is about 1% of the filn thickness, as is typical of 40 most of the filns measured, 1

30

m20

IUTaO 3/MgO/AI 2C3 -c

however, the surface loss

1000 nm wavelen gth 4.9 nm RMS surf ace rough 35 nm correlatio nlength

component is a factor of fifteen to thirty too large (greater than 1

dB/cm). Since the surface loss varies quadratically with roughness, a five-fold reduction in the roughness would make the surface losses approximately tolerable. As was stated above, the model calculations are

TEO

U) U)

0 -j1 00 0

100

200

300

400

500

Thickness (nm)

600

probably not accurate to better

than a factor of two. Whether the Figure 4. Calculated surface scattering loss versus thickness for the losses are 16 times or 25 times too large, the required 4 or 5 fold data in Figure 2. The roughness is assumed indepe ndent of reduction in roughness to obtain thickness. losses around 1 dB/cm does not change the requirement appreciably. The conclusion is that the surface roughness needs to be on the order of I nm RMS or below to provide suitably low surface loss, independent of other loss mechanisms.

194

i u RMS surface roughness vs.

i

film thickness for ferroelectric oxide thin films

i

•

LTO/MgO/Saph.

•

KNO/Saph.

CVD

LTO/Saph. CVD ×

LNO/Saph. CVD LNO/MgO Xtal

•

• D

*

o

•

×

LNO/LTO LNO/Spinel

A

l

195

As grown: 580 nm LiTaO3/MgO/AI203 60.1 nm peak-to-valley roughness 7.3 nm RMS roughness 31 dB/cm loss

After planarization: 544 nm LiTaO3IMgO/AI203 6.4 nm peak-to-valley roughness 0.4 nm RMS roughness 13 dB/cm loss

0

0

Figure 6.Atomic force micrographs of a film before and after planarization. The loss measurements on the planarized sample indicate that between one-half and twothirds of the loss is attributable to the surface roughness. Before and after planarization, there was a 20-fold decrease in roughness, which according to scattering theory corresponds to a 400-fold drop in surface scattering. The remaining loss of 13 dB/cm is primarily attributable to non-surface mechanisms. Clearly, in this sample, both surface and non-surface effects are important since eliminating either effect alone is insufficient. Indeed, it will be argued below that optical scattering at grain boundaries may be correlated with phenomena which induce surface roughness and therefore causing surface and internal losses to vary in tandem. FILM ROUGHENING DURING GROWTH There are several thin film mechanisms which can cause the roughness to increase with film thickness as was observed for certain systems in Figure 5. One likely mechanism is grain boundary grooving. We speculate that separate nuclei form on the substrate, and fail to coalesce coherently, creating grain boundaries. Once present, there is a competition between grain boundary surface energy and free surface energy.

196

The total energy is minimized by the formation of surface grooves. The volume of the film increases linearly with the groove depth, whereas the free surface area increases quadratically. The film can increase in volume without adding much additional free surface area or grain boundary surface area by forming grooves. When the grain boundary surface energy is on the same order as the free surface energy, the grooves can be an appreciable fraction of the grain size. One simple calculation assuming pyramidal grains indicates that the equilibrium groove depth Ad, is given by

Ad=(- K

(7)

where w is the grainsize, and ag and o-, are the grain boundary and free surface energies respectively. This predicts that when these conditions are met, grain growth will lead to increased grooving. The relative surface energies are unknown, as is the surface metastability, but it is plausible that in these oxide films, many dangling bonds exist, giving the grain boundaries some free-surface-like character. Grain boundary disorder which increases the surface energy and leads to grooving may reasonably be expected to produce index variation (lowering) and thus lead to both internal and surface loss. One striking example of this grooving effect is in the LiTaO3/MgO/AJ 20 3 system. Figure 7 shows an atomic force micrograph (AFM) of a LiTaO 3/MgO/Al 20 3-c film with a bivariant height distribution. The texture is bimodal as is the grain boundary angle type. The MgO buffer layer induces

Height Distribution

-

I

35 nm

5 UM

0 uM

an equal quantity of 0 and

60 degree grain boundaries in the film, as is common in LiTaO 3 and LiNbO 3 films on ( 111) MgO. 7 Equation (7) in this case would apply with two grain boundary energies and two different groove depths. Figure 7. AFM image of a LiTaO 3 MgO/A 203-c film, and height histogram.

FILM NUCLEATION AND GROWTH The planarization results suggest that optical loss can occur in comparable amounts in both surface and non-surface mechanisms. The fundamental origin and properties of the grain boundaries is therefore a logical topic for further study since producing low energy grain boundaries should result in low internal and surface loss.

197

We observe that as the film grows, the grainsize generally increases, and the grooves deepen. We have examined the initial stages of sputter growth of LiNbO 3 films on sapphire, and observed that nuclei appear separated by about 20 nm. Coherent coalescence may be prevented by the fact that the 8% lattice mismatch between c-axis LiNbO 3 and c-axis sapphire causes the lattices to fall out of phase over 1.3 nm, a distance far below the nuclei spacing. Lattice match is therefore expected to play a decisive role in the planarity of thin film systems which are susceptible to grooving. To illustrate this it is revealing to compare growths on c-plane A120 3 and MgO (111). Figures 8 and 9 show respectively TEM Figure 8 TEM cross section of a lithium niobate thin film sputtered on AI 20 3 -c. cross sections of a lithium niobate films on sapphire and MgO. Some relevant properties and measurements on the two systems are listed in Table 1. The MgO (111) basal plane has a very close lattice match (less than 0.2%) to the basal plane of lithium niobate. This compares favorably with the lattice match of lithium niobate to sapphire which is greater than 7%. Furthermore, the M oxygen sublattice of lithium niobate is Figure 9. TEM cross section of a lithium niobate film sputtered on MgO (111) hexagonal close packed, which is the same as sapphire, but distinct from the oxygen sublattice of magnesium oxide, which is face centered cubic. Comparison of lithium niobate films grown on both sapphire and magnesium oxide provides an interesting contrast between highly lattice matched heterostructural epitaxy, and poorly lattice matched isostructural epitaxy. The lithium niobate films on MgO (111) grow very planar, with surface roughness less than one nanometer, and hence meet the requirement for low optical surface scattering losses. The bimodal mix of 0 degree and 60 degree grain boundaries on MgO stems from the inequivalent structure with lithium niobate. Figure 10 shows an atomically

198

abrupt 60' boundary in the sample in Figure 9. Comparing Figures 8 and 9, the surface of the film on MgO is less grooved than on sapphire. The presence of many grain boundaries, and the lack of significant grain boundary grooving suggest coherent coalescence of the lithium niobate grains during the initial stages of growth. In fact, Figure 10 shows an atomically abrupt 60' boundary in the sample in Figure 9. This is not only an interesting possible observation of low-energy high-angle grain boundaries, but may portend low optical losses in these films. Continued effort will be directed at making samples large enough for optical characterization. The films on sapphire are monovariant and also very smooth with only small indications of grain boundary grooving.

Figure 10. TEM image of a 60 degree boundary inlithium niobate on MgO. Table I Substrate Effects on LiNbO 3 Film Properties MgO (111)

A1 20 3-c

Lattice mismatch

8%"isostructural" hcp/hcp

0.16% "heterostructural" hcp/fcc

Thermal mismatch In-plane texture

10.5 ppm/°C monovariant, low Z grain bndries

4.6 ppm/°C bivariant; 00 and 600 boundaries

Rocking width

0.0040

0.30

Grain boundaries

out-of-phase, columnar

coherent, columnar

Principal limitation

lattice, thermal mismatch

surface quality, high Z grains

CONCLUSION Optical losses limit the utility of ferroelectric film waveguides. The film thickness requirement for practical devices places a demand on film quality which is met under only limited choices of growth conditions and substrates. More specifically, practical SHG devices of 400 nm thickness require roughness on the order of I nm RMS or less. Understanding the role of grain boundaries, in their optical effects, and their effects on surface morphology appears important to growing planar thin films. True single crystal growth, free of grain boundaries, may be the most viable structure, however, this may severely limit the materials choices and growth techniques.

199

Systems with large lattice mismatch such as LiNbO 3 on sapphire may present challenges due to dense nucleation and incoherent coalescence. Systems with small lattice mismatch such as LiNbO 3 on MgO and lithium tantalate may circumvent this effect, while creating other problems. Nevertheless, the growth of lithium niobate thin filns appears to be converging on a process for viable optical waveguide devices. ACKNOWLEDGEMENTS We wish to thank Dr. Ross Bringans and Dr. James Boyce for many useful technical discussions. We thank Zihong Lu, formerly of Stanford University, Alice Chow of North Carolina State University, Huyang Xie of Cornell University and Michael Nystrom and Bruce Wessels of Northwestern University for allowing us to measure and present data on their films. This work is supported by the Department of Commerce Project ID (70NANB2H 124 1). REFERENCES 'G. F. Lipscomb, R. S. Lytel, A. J. Ticknor, J. Kenny, T. E. Van Eck, D. G. Girton, and E Binkley,_Proc. Mater. Res. Soc. Symp., 228, 15 (1992). 2 W. P. Risk, Optics and Photonics News, 1(5), 10 (1990). ' S. Nakamura, T. Mukai, and M. Senoh, Appl. Phys, Lett., 64, 1687 (1994). 4 D. K. Fork, F. Armani-Leplingard, and J. J. Kingston, proceedings of the Fall 1994 Symposium on Ferroelectric Thin Films IV, Nov. 29, San Francisco, CA. 5 K. Nashimoto, D. K. Fork, and T. H. Geballe, Appl. Phys. Lett., 60, 1199 (1992). 6W-Y Hsu, and R. Raj, Appl. Phys. Lett., 60, 3105 (1992). 7 D. K. Fork and G. B. Anderson, Appl. Phys. Lett. 63, 1029 (1993). 'L. S. Hung, J. A. Agostinelli, J. M. Mir, and L. R. Zheng, Appl. Phys. Lett., 62, 3071 (1993). 9 R. Gutmann, J. Huliger, R. Hauert, and E. M. Moser, J. Appl. Phys., 70, 2648 (1991). 'OS. Schwyn Thony, H W. Lehman, and P. Gunter, Appl. Phys. Lett., 61, 373 (1992). "T. M. Graettinger, S. H. Rou, M. S. Ameen, 0- Auciello, and A I. Kingon, Appl. Phys. Lett., 58, 1964 (1991). 12D. Marcuse, Theomy of Dielectric Optical Wiveguides, (Academic Press, New York, 1974), Chap. 3. "34D. Marcuse, Bell Sys. Tech. J., p. 3187, Dec- 1969. 1 pseudo-even and pseudo-odd designate modes which are truly odd or even in the limit of a symmetric slab guide. 15D. K. Fork, F. Armani-Leplingard, and J. J. Kingston, proceedings of the 6th International Symposium on Integrated Ferroelectrics, vol. 6, part 1, March 1994, Monterey CA. 16j_ J. Kingston et al. to be published.

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