Coupling characteristics of LiNbO3 directional couplers
Craig T. Mueller and Elsa Garmire
An analytic model of a Ti:LiNbO 3 channel waveguide has been developed to predict conveniently the cou-pling coefficient of directional couplers based on near-field intensity measurements. This model is com-pared to experimental results at wavelengths of 0.63 and 0.81 ,um and indicates an accuracy of predictingcoupling coefficients within a factor of 2 without prior knowledge of the fabrication parameters.
1. Introduction
LiNbO 3 directional couplers are being developed fora number of applications including beam splitters, highspeed modulators, and optical switches.1 In workingwith such devices it is important to understand thecoupling process and know the value of the couplingcoefficient. A number of theoretical techniques havebeen used to study the diffused directional coupler: theWKB method,2 the variational technique,3 the propa-gating-beam method,4 the transverse resonance meth-od,5 and others.6 The approach taken by researchersto date has been to use prior knowledge of the physicaldiffusion parameters to predict the coupling coefficientfrom a theoretical model of the diffusion process andcompare it with the experiment. The difficulty withthis type of approach lies in the great variability ofdiffusion conditions which makes it difficult to compareresults obtained from different groups. Furthermore,the conditions of fabrication may not even be known,so it is desirable to have a method of determining thecoupling coefficient from the physical characteristicsof the finished directional coupler.
In this paper we introduce an analytic model tocharacterize the modes of 2-D diffused channel wave-guides and compare it with experimental measurementsof the near-field intensity profiles. Section II describesthis model and its use in predicting normalized wave-guide parameters for our channel waveguides. We use
When this work was done both authors were with University ofSouthern California, Center for Laser Studies, Los Angeles, California90089-1112; C. T. Mueller is now with Aerospace Corporation, Elec-tronics Research Laboratory, P.O. Box 92957, Los Angeles, California90009.
measurements at two wavelengths of the channelwaveguide mode intensity depth and width to deter-mine the maximum index change An, the diffusiondepth D, and the halfwidth of the index profile W. InSec. III additional support for this model is found whentitanium (Ti) diffusion parameters, specifically thediffusion constant and the change of the refractive indexwith Ti concentration, inferred from these measure-ments are seen to be in good agreement with previouslypublished values.7 Using the predicted channelwaveguide mode parameters, the coupling coefficientis calculated for a diffused directional coupler in Sec.IV, and the coupling length is plotted as a function ofwaveguide separation with no free parameters. Com-parison of the model to experimental measurements ondirectional couplers, which are single mode from 0.63to 0.85 jAm, indicates that the coupling lengths arepredicted within a factor of 2 over this wavelengthrange.
11. Mode Profile
A. Modeling
To calculate the coupling coefficient, which dependson the overlap of the waveguide modes, a model of thediffused channel waveguide modes must be developed.To simplify the analysis and limit the number of nu-merical calculations, we sought index profile functionswhich would produce analytic field solutions of the waveequation. Consequently, the exponential and sech2
index profiles were chosen to approximate the indexprofile perpendicular and parallel to the crystal surface,respectively. Comparison to the real solutions of thediffusion problem and direct measurements of the Ticoncentration profile support the choice of these par-ticular diffusion profiles.8 Constructing equivalentplanar waveguides with these index profiles, the effec-tive index method for diffused channel waveguides9 wasused to calculate the mode dispersion parameters.
The geometry for the directional coupler is shown inFig. 1(a). Figure 1(b) diagrams the way in which the
effective index method is used. Following the effectiveindex method used for characterizing diffused channelwaveguides,9 the channel waveguide consists of anequivalent planar waveguide parallel to the crystalsurface G and an equivalent planar waveguide parallelto the normal to the crystal surface G'. The formerwaveguide has normalized waveguide parameters V andb, while the latter has normalized waveguide parametersV' and b'. V and V', the normalized diffusion lengths,and b and b', the normalized mode indices, have beendefined in terms of three physical parameters, the re-fractive-index change at the waveguide surface An, thediffusion depth D, and the diffusion width W to be
V = kD(2nb An)1/ 2 for An << nb, (1)
b = (neff - nb)/(An) for An << nb, (2)
V'= Vbl/ 22W/D, (3)
b' = (flff - nb)/(neff - nb) for An << nb, (4)
where k = 2ir/X = propagation constant,A = wavelength,
nb = refractive index of the bulk mate-rial,
neff = mode effective index (= /k).Note that
neff = nb + Anbb' for An << nb. (5)
Beginning with the waveguide parallel to the crystalsurface G, an exponential index profile was chosen be-cause it has analytic field solutions and was found toproduce mode intensity profiles closely matching theexperimental data. The exponential index profile andcorresponding electric field arel 0
n(x) = nb + An exp(-y/D) rory > 0, (6)
E. (y) = AJU exp(-y/2D)], y > 0 in LiNbO3, (7)
E. (y) = AJu[4I exp(+vy/2D), y S 0 in the air, (8)
where J = Bessel function of order u;u = 2kD(n2ff- n2)1/2 = 2b1/2V;
The corresponding dispersion relation for the lowest-order mode in waveguides of this index profile is
-/i7Th- V tanlv/ -1 = 3ir/(8V). (9)
For the waveguide in a plane normal to the surface G',we then assumed an index profile with a sech2 functionaldependence:
n(x) = nb + An sech2(x/W), (10)
where W is the parameter measuring the halfwidth ofthe index profile. The solutions of this index profilehave also been shown to be analytic and are of the fol-lowing form for the lowest-order model':
E = Asech(x/W)I(V'v-1/
where A is a constant, and its dispersion relation is
[(V2 + 1)1/2 1]2V(
11)
lal
T _X ; X X
D D~~~6 G G'
Ib
Fig. 1. (a) Geometry for the directional coupler interaction region.(b) Effective index method used to separate the channel waveguideinto equivalent planar waveguides, G and G', parallel to thecrystal surface and parallel to the normal to the crystal surface,
respectively.
B. Experiments
Directional couplers were fabricated in z-cut LiNbO3crystal. The individual channel waveguides were 3 jimwide, extending along a direction parallel to the y axisof the crystal. Photomasks which formed the direc-tional coupler structure were made by double exposinga mask containing a single-channel waveguide bend.1 2
Parallel directional couplers were photolithographicallydefined with center-to-center separations ranging from6 to 10 jim and a constant interaction length of 1.5 mm.Waveguides were formed by evaporating a 520-A layerof Ti on top of the LiNbO3 crystal, placing it in a dif-fusion oven, and diffusing for 4 h at 10000 C in a flowingargon atmosphere, followed by a cool-down inoxygen.
Measurements of the near-field intensity profile weremade by imaging the output from the polished wave-guide edge with a 40X microscope objective (N.A. 0.55)onto a 25-jim pinhole at a distance of -1 m from thelens. The resolution was limited by the diffraction limitof the objective lens to be -1 Aim. A magnification of219X was determined by measuring the separation ofthe two outputs of the directional coupler at the scan-ning pinhole and dividing by that measured under amicroscope at the crystal edge. A number of scans,fourteen to be exact, parallel to the crystal surface weretaken at different positions along the normal to thatsurface, and the intensity data from the detector wererecorded on a digital storage scope for further process-ing. From these data the intensity profiles both alongthe crystal surface and in a direction normal to thecrystal surface were determined.
C. Comparison of Theory and Experiments
A comparison between theory and experiment can bemade since measurements of the intensity FWHMalong x and y at two wavelengths give four independent
(12) data points. By using the effective index method, we
are able to characterize these channel waveguides withonly three free parameters: An, D, and W. Hence thefourth measurement provides a check on the accuracyof the model, which was determined to be within 20%when compared with the experimental data.
Figure 2 shows the comparison of the calculated (solidline) and experimental intensity profiles along the di-rection normal to the crystal surface for X = 0.63 im.The dashed line represents a convolution of the calcu-lated intensity profile and the optical transfer functionof the imaging system, accounting for the limited reso-lution of the objective lens. The waveguide parametersAn 0.017 and D = 1.0 jim were determined by theo-retical fits of intensity profiles to experimental FWHMdata points at wavelengths of 0.63 and 0.81 jim. Im-posing the additional condition that the waveguideparameters remain in the single-mode regime limitedthe range of the V parameter to 37r/8 < V < 7-r/8. Thewaveguide parameters shown in Fig. 1 were found bycomparing the FWHM from a number of An and Dvalues to those measured at 0.63 and 0.81 jim. Fol-lowing this procedure, the accuracy of the waveguideparameters, An and D, was determined to be 10%.
The effective index method was used to determinethe waveguide parameters for the equivalent waveguideparallel to the normal of the crystal surface. Insertingthe values of An = 0.017 and D = 1.0 jim determinedfrom the intensity profile measurements along thenormal to the crystal surface into Eqs. (1)-(5), V' waspredicted to be 2.77 at X = 0.63 jim. Since An is alreadydetermined, there is really only one free parameter W,which determines the waveguide characteristics in thisplane. The sech2 index profile does, however, requirethat 0 < V' < 2J for the waveguide to exhibit single-mode behavior. Using this requirement, W was fittedby minimizing the sum of the squares of the deviationof the calculated FWHM from the measured FWHMof 2.1 and 2.7 jim at X = 0.63 and 0.81 jim, respectively.As a result, W was set equal to 1.2 ,um, and this valuewas used to calculate the intensity profile shown withthe experimental data in Fig. 3. It is clear that thecalculated intensity profile fits closely the experimentaldata except in the tails where the intensity predicted bythe sech2 index model falls off slower than the measuredprofile.
Our measurement of the in-plane and out-of-planeintensity FWHM at two different wavelengths givesfour independent pieces of data. If we use these datato determine the maximum refractive-index increaseAn, the waveguide depth D, and width W, we have usedthree pieces of data to determine the three parameters.These three parameters may be inserted into the modelto predict the fourth piece of data, the in-plane intensityFWHM at 0.81 jim. It is clear that the best accuracyis obtained when the wavelength difference between themeasurements is maximized. On the other hand, toavoid discriminating between multiple modes whenmeasuring the FWHM depth and width, the lowerwavelength should be chosen so that V' is below cutofffor all higher-order modes. For these particularchannel waveguides, the model predicts single-mode
-1 0 1 2 3 4 5 6 7
y, m
Fig. 2. Intensity profile parallel to the normal to the crystal surface(X = 0.63 Am).
1.0r
0.51
-4 -3 -2 -I 0 1 2 3 4x, Am
Fig. 3. Intensity profile parallel to the crystal surface (X = 0.63pm).
behavior for X > 0.62 jim. Ideally, a tunable wavelengthlaser source is required to maximize the wavelengthdifference between measurements and, therefore, obtainthe highest accuracy. However, using available lasersources we were able to predict theoretically the in-plane FWHM from An, D, and W to within 20% of theexperimentally measured FWHM of 2.7 jim.
Ill. Ti-Diffusion Parameters
Two physical parameters of importance to the fab-rication of Ti-indiffused LiNbO3 waveguides are thediffusion constant and the change of the refractive indexwith Ti concentration, dno/dC. The concentrationprofiles for this type of diffusion process are theoreti-cally predicted to be error function complement orGaussian in the limits where the source can be consid-ered infinite or finite, respectively. In this work wehave chosen to use an exponential diffusion profile toobecause of its analytic field solutions. The lie diffusiondepth D and maximum index change were predictedearlier to be 1 jAm and 0.017 by matching experimentalnear-field intensity profiles to this profile at twowavelengths. Based on previous work13 the physicaldiffusion profile is expected to be erfc with a corre-sponding diffusion length defined by
DI = 2(Zt) 1/2, (13)
where t is the diffusion time. To equate the two profilesthe diffusion length for the erfc profile was set equal to1.8 jim. In this way, the error from overestimating theconcentration in the tails of the exponential functionis approximately offset by underestimating it at smally values.
Inserting the diffusion length of 1.8 jim and the ex-perimental diffusion time at 10000C of 4 h for thissample into Eq. (13), the diffusion coefficient along thez axis of the crystal for z -cut diffusion is predicted to beZ, = 5.6 X 10-13 cm2/sec. Burns et al. 14 reported fZ
= 1.4 X 10-13 cm2/sec obtained from matching mea-sured mode effective indices of multimode waveguidesto normalized dispersion curves assuming a Gaussianindex profile. The discrepancy in these two measure-ments may be attributable to variations in the LiNbO3composition of the substrates' 5 or differences in thefabrication process. To put this in perspective it isuseful to examine the experimental results on the dif-fusion coefficient measured along the y crystallographicdirection at 1000°C. Previously Mueller et al. 13 mea-sured Dy to be 5.0 X 10-13 cm2 /sec. Similarly, Fukumaet al.,7 using an electron microprobing technique,measured Dy to be 4.6 X 10-13 cm2/sec. By contrast,the samples measured by Burns et al. 13 gave ,y = 9.4X 10-13 cm2/sec, almost a factor of 2 higher. Further-more, Holmes and Smyth15 have suggested that no an-isotropy exists in the diffusion coefficient with crystalorientation and have found it vary by an order of mag-nitude for samples having different Li2O mol % com-positions. However, for a congruent composition (48.6mol % Li2O), their data yield O = 4.8 X 10-13 cm2/secat 10000C, which compares favorably with our result.
The second parameter, the change of the refractiveindex with Ti concentration (i.e., dno/dC), is derivedfrom the relationship between the surface index changeAn and the concentration of Ti at the surface C(0) (Ref.16):
An = C(O)dno/dC. (14)
Assuming the titanium concentration in the waveguideto be exponential and applying the conservation of ti-tanium, we set the integral of the titanium concentra-tion in the waveguide equal to the product of the densityand thickness of the titanium film prior to diffusion asfollows:
C (O) exp(-y'/D)dy' = at. (15)
Solving Eq. (15) gives C(0) = axt/D. Substituting thisresult into Eq. (14), we now have an expression for thechange of the refractive index with Ti concentration:
dno/dC = AnD/(at). (16)
Inserting the values found in Sec. II for the exponentialprofile An = 0.017 and D = 1.0 jm along with the tita-
V'2
K =
8w b'V,2 + 16x 2W2n2]112
nium density, at = 5.71 X 1022 cm 3 ,17 and the measuredfilm thickness t = 520 d 20 A, our model predictsdno/dC = 5.7 X 10-24 cm3 . Minakata et al.'8 havemeasured dno/dC = 5.6 X 10-24 cm3 using optical in-terferometry, and Burns et al. 14 have reported dno/dC
= 9.0 X 10-24 cm3 . Again, the factor of 2 between thenumbers obtained by the author (and Minakata et al.)compared with Burns et al. indicates possible differ-ences in fabrication techniques and/or substrates.
IV. Coupling Coefficient
The directional couplers studied here are of the uni-form spacing type in which the coupling coefficient re-mains constant along the interaction region. The op-tical field profiles and the dispersion characteristics ofthe channel waveguides are assumed to be identical tothose of the model discussed in Sec. II. To apply theseresults to analysis of diffused directional couplers, wechose to calculate the coupling coefficient using a cou-pled-mode theory approach analogous to that which hasbeen done for the step-index case.'9 Then, by insertingthe results of the model developed for a single diffusedchannel waveguide into this equation, an expression forthe coupling coefficient is derived which can be easilycalculated numerically on a desktop computer.
The first step in calculation of the coupling coefficientis to write out the refractive-index distribution of thedirectional coupler structure in the interaction region.Using the assumption of a sech2 profile parallel to thesurface of the crystal and the approximation that An <<nb the square of the refractive index is
n 2 (x) = nb + 2nbAn[g(x + d/2) + g(x -d/2)], (17)
where g is the function describing the diffusion profile.Approximating the total electric and magnetic fields ofthe modes by a linear superposition of the fields of thetwo adjacent waveguide modes, substituting intoMaxwell's equations, multiplying by the complex con-jugate of the time-dependent terms, and solving the twoequations to first order, the coupling coefficient hasbeen shown to be'9
k2 nbAn JbO2f3
g(x + d/2)El E2dx
f E.1 Eldx
, (18)
where E1,2 is the electric field in guides 1 and 2, re-spectively.
In our case, we will assume that each channel wave-guide has identical waveguiding properties and is loss-less. Substituting the index difference term, the indexprofile, and the electric field of a single-channel wave-guide from Eqs. (10), (11), and (17), respectively, thecoupling coefficient becomes
sechvv'/22 x + d/2) h7V/2 (x ) d(1
E sechj/WVV'x +) dxBecause of the complicated nature of the integrals in thenumerator and denominator, analytic solutions couldnot be found, and both integrals were calculated nu-merically using Simpson's method on a desktop com-puter. It is the relatively simple form of these integrals
Fig. 4. Coupling length as a function of waveguide separation.
which is the advantage of the analytic model over a fullnumerical analysis. Last, the dispersion of the LiNbO 3crystal 2 0 was also accounted for in these calculations.
Figure 4 shows a comparison of the calculated andmeasured coupling lengths, 1(1 = 7r/2K), as a function ofcenter-to-center waveguide spacing d for three wave-lengths. The calculated curves were obtained from thewaveguide parameters determined by the near-fieldintensity fit at X = 0.63 jim as described in Sec. II. Notethat the diffused waveguide model for calculating thecoupling coefficient predicts the experimental couplinglengths within factors of 2 for the two wavelengths aswell as the exponential dependence on waveguide sep-aration over a range of a factor of 20 in coupling lengthwithout the introduction of free parameters. Thesmaller slope of the calculated coupling lengths de-pendence on waveguide separation at 0.81 jim is mostlikely due to the slower falloff of the fields in the sech2
index model when compared to the actual measuredmode profiles as shown in Fig. 3.
V. Conclusions
An analytic model has been developed to study themode profiles of Ti-indiffused LiNbO3 channel wave-guides and to predict the coupling coefficient (or cou-pling length) of directional couplers as a function ofcenter-to-center waveguide spacing for several wave-lengths. Comparison with single-mode directionalcouplers indicates that the FWHM of the near-fieldintensity profile at a wavelength of 0.81 m is predict-able from data taken at 0.63 jim with an accuracy of20%. Based on near-field intensity measurements, thecoupling coefficients of directional couplers have beenpredicted within factors of 2 with no free parametersover wavelengths ranging from 0.63 to 0.85 jim. Theadvantages of this technique are that prior knowledgeof the diffusion conditions or parameters is not required,and the equations are sufficiently simple to allow therequired calculations to be done on a desktop com-puter.
We would like to thank L. Johnson for the loan of themask used for these experiments.
References1. R. C. Alferness, "Guided-Wave Devices for Optical Communi-
cation," IEEE J. Quantum Electron. QE-17, 946 (1981).2. T. Suhara, Y. Handa, H. Nishihara, and J. Koyama, "Analysis
of Optical Channel Waveguides and Directional Couplers withGraded-Index Profile," J. Opt. Soc. Am. 69, 807 (1979).
3. A. Sharma, E. Sharma, I. C. Goyal, and A. K. Ghatak, "TheVariational Technique for Diffused Directional Couplers," Opt.Commun. 34, 39 (1980).
4. M. D. Feit, J. A. Fleck, Jr., and L. McCaughan, "Comparison ofCalculated and Measured Performance of Diffused Channel-Waveguide Couplers," J. Opt. Soc. Am. 73, 1296 (1983).
5. J. Ctyroky, M. Hofman, J. Janta, and J. Schrofel, "3-D Analysisof LiNbO 3 : Ti Channel Waveguides and Directional Couplers,"IEEE J. Quantum Electron. QE-20, 400 (1984).
6. C. H. Bulmer and W. K. Burns, "Polarization Characteristics ofLiNbO 3 Channel Waveguide Directional Couplers," IEEE/OSAJ. Lightwave Technol. LT-1, 227 (1983).
7. M. Fukuma, J. Noda, and H. Iwasaki, "Optical Properties in Ti-tanium-Diffused LiNbO3 Strip Waveguides," J. Appl. Phys. 49,3693 (1978).
8. C. T. Mueller, Ph.D. Dissertation, U. Southern California(1983).
9. See, for example, G. B. Hocker and W. K. Burns, "Mode Dis-persion in Diffused Channel Waveguides by the Effective IndexMethod," Appl. Opt. 16, 113 (1977).
10. E. M. Conwell, "Modes in Optical Waveguides Formed by Dif-fusion," Appl. Phys. Lett. 23, 328 (1973).
11. D. F. Nelson and V. J. McKenna, "Electromagnetic Modes ofAnisotropic Dielectric Waveguides at p-n Junctions," J. Appl.Phys. 38, 4057 (1967).
12. L. M. Johnson and F. J. Leonberger, "Low-loss LiNbO 3 Wave-guide Bends with Coherent Coupling," Opt. Lett. 8, 111(1983).
13. C. T. Mueller, C. T. Sullivan, W. S. C. Chang, D. G. Hall, J. D.Zino, and R. R. Rice, "An Analysis of the Coupling of an InjectionLaser Diode to a Planar LiNbO 3 Waveguide," IEEE J. QuantumElectron. QE-16, 363 (1980).
14. W. K. Burns, P. H. Klein, and E. J. West, "Ti Diffusion in Ti:LiNbO 3 Planar and Channel Optical Waveguides," J. Appl. Phys.50, 6175 (1979).
15. R. J. Holmes and D. M. Smyth, "Titanium Diffusion into LiNbO 3as a Function of Stoichiometry," J. Appl. Phys. 55, 3531(1984).
16. R. V. Schmidt and I. P. Kaminow, "Metal-Diffused OpticalWaveguides in LiNbO 3 ," Appl. Phys. Lett. 25, 458 (1974).
17. Handbook of Chemistry and Physics (CRC Press, Cleveland,1984-85).
18. M. Minakata, S. Saito, M. Shibata, and S. Miyazawa, "PreciseDetermination of Refractive-Index Changes in Ti-DiffusedLiNbO 3 Optical Waveguides," J. Appl. Phys. 49, 4677 (1978).
19. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold,New York, 1982).
20. M. V. Hobden, and J. Warner, "The Temperature Dependenceof the Refractive Indices of Pure Lithium Niobate," Phys. Lett.22, 243 (1966).
