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Kulishov et al. Vol. 18, No. 4 /April 2001 /J. Opt. Soc. Am. B 457Varennas, Quebec J3X 1S2, Canada

Received April 25, 2000; revised manuscript received November 16, 2000

A new design for an electro-optically induced tilted phase grating inside a waveguide is proposed. The electricfield and the refractive-index distribution induced inside a waveguide by voltage applied to two systems ofinterdigitated electrodes that are shifted with respect to each other are calculated rigorously on the basis of anoriginal technique. The model accounts for the arbitrary electrode shift distance d (0 < d < 2l), where l isthe electrode spatial period. It is shown that the proper choice of the shift can minimize the structures ca-pacitance and consequently its time response. The refractive-index distributions are calculated for variousschemes for application of electric potential and electrode position that demonstrate the possibility of switchingthe direction of the grating wave vector. It is shown how the concept can be use to build electro-opticallycontrollable transmissive (long-period) and reflective (short-period) tilted gratings and couplers in both multi-layered (transverse) and planar (lateral) configurations. 2001 Optical Society of America

OCIS codes: 130.2790, 050.1950, 060.1810, 160.2100, 130.3120.

1. INTRODUCTIONWavelength-division multiplexing devices are highly use-ful in increasing the information-carrying capacity of op-tical fiber networks. Todays multiplexingdemultiplexing schemes involve fixed-wavelength filters,such as in-fiber Bragg gratings, integrated waveguidegratings, and arrayed waveguide gratings. Tunable-filter devices that will be able to select dynamically an in-dividual channel for a stream of data are needed in recon-figurable optical networks, and the addition of tunable-filter technologies can be expected in the near future.The new tunable devices will most probably combine dif-fraction grating or arrayed waveguide grating conceptswith active-device geometries. Examples that employlong-period fiber gratings might include devices based onthe interaction of intentionally excited fiber claddingmodes with the fibers surrounding material as well as ad-justable spectral filters for dynamic gain flattening. Fur-thermore, fiber Bragg gratings integrated into an exter-nally controllable geometry could form the basis forprogrammable optical devices with refractive-index dis-

Developing new tunable-grating devices requires pow-erful computer simulation tools that will be able to relatethe parameters of the structure, including material andgeometrical parameters, in the process of device optimiza-tion. An integral part of the simulation software tool isan electrostatic problem solver that is able to calculatethe electric field distribution as well as the resultantelectro-optically (EO) induced refractive-index change inthe waveguide or cladding parts of the device. In this pa-per we present a flexible and versatile technique for cal-culation of electric field and induced refractive-indexmodulation that includes a number of waveguide param-eters. Particular emphasis is given to analysis of devicesthat include EO induced tilted gratings. Our analysis fo-cuses on calculation of refractive-index distribution andthe structure capacitance, the latter determining the timeresponse and the switching energy budget. Opticalanalysis of guided-wave interaction in reflection andtransmission is beyond the scope of this paper. We in-tend to address the wave-coupling problem based on solu-tion of the electro-optical problem described in this paperElectro-optically inducin wav

Mykola

Adtek Photomask, Inc., 4950 Fisher S

Pavel

Institute for National Measurement StandardsOntario K1A

Xavier

Departement de Genie Physique et Genie des MaterStation Centreville, Institut National de la Recherc

Sebastie

Institute National de la Recherche Scientifique, (INRtributions that are dynamically controllable in accordancewith network operating conditions.1

0740-3224/2001/040457-08$15.00 tilted phase gratingsuides

lishov

t, Montreal, Quebec H4T 1J6, Canada

eben

ational Research Council of Canada, Ottawa,6, Canada

xhelet

, Ecole Polytechnique de Montreal, P.O. Box 6079,Scientifique, Montreal, Quebec H3C 3A7, Canada

elprat

nergie et Materiaux), 1650 Boulevard Lionel-Boulet,in a future publication.Tilted waveguide and fiber phase gratings have an im-

2001 Optical Society of America

458 J. Opt. Soc. Am. B/Vol. 18, No. 4 /April 2001 Kulishov et al.portant role in photonics devices. Their ability to providecoupling between guided mode(s) and radiation mode(s) istaken as a principle of operation for waveguidefibertaps, broadband filters, inputoutput couplers, spectrumanalyzers, and so on. Traditionally these unsymmetricalgratings are manufactured in the prefixed shape of a cor-rugated waveguide surface through, for example,reactive-ion etching or, in the case of tilted fiber grating,by ultraviolet light writing into the core of a fiber; i.e., theoptical characteristics of these unsymmetrical gratingscannot be changed.

An EO induced grating can be an important buildingblock for modern optical networking and information-processing technologies that require high degrees of par-allelism, high switching speed, and versatility. Theelectro-optical phase gratings have attracted significantresearch attention because of the important applicationsof these gratings in optical interconnection,2

communication,3 memory, and computing.4 EO inducedgratings are dynamic phase gratings; thus they can betuned or switched on and off and have a generally lowmodulation level of refractive index owing to their appliedelectric fields. The periodic modulation is crated by aninterdigitated electrode (IDE) structure. The electrodescan be either a bipolar counterdirectional couple5 or asubstrate plane electrode combined with a single comb-like cladding electrode. The former structure requireslower voltages; the latter allows for shorter periods withan additional freedom to EO change both constant andvariable components of the electric field. The constantcomponent can be used to control the average value of thewaveguide refractive index and in this way tune the peakreflectivity. This concept was elaborated in a recentpublication.6 However, the geometries published so farsuppose distributions in which the EO periodic refractiveindex is induced symmetrically with respect to the propa-gation axis. In this paper we propose the design of awaveguide with a periodic electrode structure that per-mits unsymmetrical index distributions such as an EOcontrollable tilted grating.

EO controllable waveguide phase gratings are particu-larly attractive because of rapid advances in nanoscalefabrication that make IDE structures with submicrome-ter spacing feasible.7,8

2. CALCULATION METHODA schematic cross-sectional view of the problem to be ana-lyzed is shown in Fig. 1, where 2h and d are the distance

Fig. 1. Cross-sectional view of the waveguide electro-optic grat-ing.and the relative shift between the top and the bottomelectrode structures, respectively, l is the electrodes spa-tial period, and a is the electrodes width. The differencein dielectric properties between those of the core andthose of cladding is normally so small that it can be ne-glected in the solution of the electrostatic problem if onewishes not to be overloaded with cumbersome mathemat-ics. The electrodes on both sides of the waveguide are as-sumed to be infinitely thin. This approximation is justi-fied because the electrodes thickness is much less thanother dimensions of the structure, the dielectric constantof the cladding is rather high, and the electric charge inthe electrodes is distributed mostly on the interface be-tween the electrodes and the cladding material. In theelectrostatic problem, movement of the electric chargedoes not contribute to the electric field distribution.Therefore the electrodes can be considered perfectly con-ducting. The EO guiding slab is assumed to be crystal-line with the principal dielectric axes aligned along the xand the z directions, so the dielectric properties can be de-scribed by two diagonal relative permittivities, e11 ande33 , where e is the relative dielectric constant of the sur-rounding medium. The electric potential must be a solu-tion of Laplacs equation; taking into account the problemsymmetry and the standard boundary conditions, one canwrite the potential for z > 1h and 2h < z < 1h as thefollowing expansion of harmonic and hyperbolic functions:

w~1 !~x, z ! 5 V0(n50

An exp@2~n 1 1/2!kz#

3 cos@~n 1 1/2!x#, (1)

w~2 !~x, z ! 5 V0(n50

En sinh@~n 1 1/2!kd ~z 1 h !#

3 cos@~n 1 1/2!x#

1 Dn sinh@~n 1 1/2!kd ~h 2 z !#

3 cos@~n 1 1/2!~ x 2 kd !#, (2)

where d 5 (e11 /e33)1/2, k 5 2p/l, x 5 kx, and An , En ,

and Dn are the Fourier coefficients to be found. From theboundary conditions w (2)(x 2 d,1h) w (2)(x,2h) wecan find that En 5 Dn , and, using the condition w

(1)(x,1h) 5 w (2)(x,1h), we can exclude An . Then the bound-ary condition for the surface potential and the normalcomponent of the electric field displacement can be writ-ten in the following form:

(n50

En*Hn cos@~n 1 1/2!x ! 5 1, 0 < x < a,

(n50

~n 1 1/2!En*$cos@~n 1 1/2!x# 2 Gn

3 cos@~n 1 1/2!~ x 2 kd !#% 5 0, a < x < p, (3)

where a 5 pa/l and

Kulishov et al. Vol. 18, No. 4 /April 2001 /J. Opt. Soc. Am. B 459Gn 5 H cosh@2khd ~n 1 1/2!#1

e

e33dsinh@2khd ~n 1 1/2!#J 21,

Hn 5 sinh@2khd ~n 1 1/2!#Gn , En* 5 En /Gn . (4)

To provide better convergence for the calculationmethod, it is convenient to rewrite Eqs. (3) as follows:

(n50

En*~1 1 Fn!cos@~n 1 1/2!x !

5e33d

e33d 1 e, 0 < x < a, (5a)

(n50

~n 1 1/2!En*$cos@~n 1 1/2!x#

2 Gncos@~n 1 1/2!~ x 2 kd !#% 5 0,

a < x < p, (5b)

where Fn 5 (e33d 1 e)Hn /(e33d) 2 1 and limn Fn5 0. Integrating Eq. (5b) with respect to x from x . ato p, we get

(n50

En*$sin@~n 1 1/2!x# 2 Gn sin@~n 1 1/2!

3 ~ x 2 kd !]% 5 C, a < x < p, (6)

where we can find C by setting x 5 p:

C 5 (n50

~ 2 1 !nEn*$1 1 Gn cos@~n 1 1/2!kd#%. (7)

Multiplying Eq. (5a) by (A2/p)(cos x 2 cos j) 2 1/2 and Eq.(6) by (A2/p)(cos j 2 cos x) 2 1/2, integrating the firstequation with respect to x from 0 to j , a and the secondone with respect to x from j . a to p, and taking into ac-count the known expressions for complete elliptic inte-grals of the first kind:

2

pKS sin j2 D 5 A2p E0j dx~cos x 2 cos j!1/2 ,

2

pKS cos j2 D 5 A2p Ejp dx~cos j 2 cos x !1/2 , (8)

we get the following system of equations:

(n50

En*Pn~cos j!

52

p

e33d

e33d 1 eKS sin j2 D 2 (n50

En*FnPn~cos j!,

0 < j#a;(n50

En*Pn~cos j!

52

pCKS cos j2 D 2 (n50

En*GnQn~j!,

a , j < p, (9)

where Pn(cos j) are Legendre polynomials and

Qn~j! 5A2pE

j

p sin@~n 1 1/2!~ x 2 kd !#dx

~cos j 2 cos x !1/2. (10)

Using the polynomials orthogonality condition:

E0

p

Pn~cos j!Pk~cos j!sin jdj 5 H 0 n k1k 1 1/2

n 5 k

,

(11)

we transform Eqs. (9) into the infinite set of algebraicequations with which to calculate En*:

(m50

H amk Fm 1 bmkGm 1 dmkm 1 1/2 2 2p ~ 2 1 !kNm3 ~1 1 Gm!cos@~m 1 1/2!kd#J Em*

52p

e 1 e33d

e33dMk , (12)

where

Fig. 2. Normalized electric potential distribution inside theelectro-optic slab for a/l 5 0.6, 2h 5 34 l, d 5

34 l, and a

LiNbO3 wafer (e11 5 85, e33 5 29), e 5 1.

Fig. 3. Normalized surface charge distribution for 2h 5 34 l, d

534 l, and LiNbO3 wafer (e11 5 85, e33 5 29). e 5 1 and a/l

5 0.5 (solid curve) and a/l 5 0.25 (dashed curve).

460 J. Opt. Soc. Am. B/Vol. 18, No. 4 /April 2001 Kulishov et al.amk 5 E0

aPm~cos j!Pk~cos j!sin jdj,

bmk 5 Ea

p

Pm~cos j!Qk~j!sin jdj,

Mk 5 E0

aKS sin j2 DPk~cos j!sin jdj,

Nk 5 Ea

p

KS cos j2 DPk~cos j!sin jdj. (13)We solved system (12) numerically by truncating it after afinite number of equations. We used the program Math-cad Plus, V. 6.0, from MathSoft, Inc., to write the solutionand carry out all the modeling. The solution method pro-vides good convergence as one can see from Figs. 2 and 3,were three-dimensional normalized potential distributionw (2)/V0 [Eq. (2)] and surface charge density r(x, z5 h)h/V0 are shown for truncated series after n 5 10.

As we can see from the solution, the electric field for agiven configuration of applied voltage has only a variablecomponent with a periodicity of the fundamental har-monic equal to electrode period 2l.

3. PSEUDOCAPACITANCE OF THESTRUCTUREThe main reason for limited practical applications of theEO induced gratings is that, with a relatively weak phasegrating, effective coupling needs a long coupling length,i.e., larger electrode regions that limit the time responseand therefore the upper frequency. In a previouspublication9 it was shown that a double-sided electrodestructure with a half-period shift between these sides pro-vides a lower capacitance value than does the traditionalone-sided IDE structure for 2h . 0.23l. Therefore it isof interest to show how capacitance will depend on shift dfor the structure under consideration. Equation (5b) de-scribes, within a constant factor V0(e33e11)

1/2/h, the den-sity of the electric charge distribution the electrodes. In-tegrating the charge distribution over an electrode stripe2 pa/l < x < 1pa/l, we can get a total charge per elec-trode stripe unit length:

Q 5V0~e11e33!

1/2

h (n50

En* sin@~n 1 1/2!ka/2#

3 $1 2 Gn cos@~n 1 1/2!kd#%. (14)

This expression gives us the opportunity to estimate thestructure capacitance by using the pseudocapacitance(PC) approach,10 which is defined as the situation when aratio of the total charge per unit length is brought abouton the electrode stripe to the potential on the stripe, C5 Q/V0 . In Fig. 4, the plots of the dependence of the PCon the electrode shift are presented for different valuesa/l and 2h/l ratios of the electrode duty. As we can see,the PC curves achieve their minimum and maximum val-ues depending on the value of electrode shift d. The lessthe distance between the top and the bottom electrodes is,the more marked is the difference between minimum andmaximum values; for 2h > l this difference is becomingbarely noticeable [Fig. 4(c)]. This result confirms the factthat an electric field excited by a patterned electrode de-cays rapidly as its distance from the electrodes increases;the pattern is essentially washed out at a distance fromthe electrodes that is equal to the patterns feature size.Here, using a double-sided structure, we at least doublethis distance; to achieve good penetration of the field intothe waveguide one should keep the distance between thetop and the bottom electrode, 2h, less than electrode pe-riod l. However, for these 2h/l values the PC stronglydepends on the electrode duty ratio, with increasing curveasymmetry for narrow electrodes [Fig. 4(a)]. If for a/l5 0.6 the PC distribution is nearly symmetric, reachingits maximum value for d l and its minimum for d5 0, for a/l 5 0.35 the maximum is at d 0.7l, and theminimum is at d 1.7l. Comparison of Figs. 4(a) and4(b) shows us that the difference between minimum PCvalues for various duty ratios is increasing as the distancebetween the top and the bottom electrodes increases.Another interesting result of PC behavior is that, for 1.5& d/l & 2, proximity of the top and bottom electrodes ac-tually reduces the PC value [Fig. 4(c)].

Fig. 4. Electrode PC as a function of the relative shift betweenthe top and the bottom electrodes for several values of the elec-trode duty ratio: (a/l 5 0.35, dotted curves; a/l 5 0.5, solidcurves; a/l 5 0.6, dashed curves) and for several valuesof the electrode-distance-to-electrode-period ratio: (2h/l 5 0.5,crosses; 2h/l 5 0.75, diamonds; and 2h/l 5 1, squares). Thecalculation was made for LiNbO3 wafer (e11 5 85, e33 5 29),e 5 1.

Kulishov et al. Vol. 18, No. 4 /April 2001 /J. Opt. Soc. Am. B 4614. TRANSMISSIVE TILTED ELECTRO-OPTICGRATINGSThe asymmetry of the electric field is used in the design toinduce a tilted grating. If we take an electro-optic mate-rial (crystal or polymer) with a 4mm or a 3m crystallo-graphic point group, for example, LiNbO3, BaTiO3, andhigh-mb (molecular dipole momentum and hyperpolariz-ability) chromophores,11,12 the electro-optic coefficient r33can be utilized when the guided mode is polarized alongthe optical axis and an electric field lies along the opticalaxis. For the design under consideration (Fig. 1) TM-likemodes have to be utilized to produce the highest modula-tion effect.6 For this condition the refractive-index incre-ment is Dn 1/2ne

3r33Ez , where ne is the extraordinaryrefractive index and Ez 5 2]w(x,z)/]z is the normalcomponent of the electric field inside the waveguide. Fig-ure 5 presents contour plots of the refractive-index distri-bution for d 5 1.7l (low PC state) and d 5 0.7l (high PCstate), which one can attain simply by shifting the voltagesequence between the top and bottom electrodes. The tiltangle is defined by the distance between the top and bot-tom electrodes, 2h, and their shift d:

q 5 5 arctanSl 2 d

2h D 0 < d < l2 arctanS l 1 d2h D l , d < 2l

. (15)

Fig. 5. Contour plots of the refractive-index distribution insidethe waveguide for (a) d 5 1.7l and (b) d 5 0.7l. The calcula-tion was made for LiNbO3 (e11 5 85, e33 5 29), e 5 1, 2h5 0.5l, V0 5 3V, h 5 1 mm; ne 5 2.3, r33 5 30 3 10

2 12

m/V, and a/l 5 0.5. The induced refractive index varies as2.29805 < n < 2.30195.For Fig. 5, where 2h/l 5 0.5, the shifting voltage resultsin the tilts switching between q1 5 34 and q25 260.5; for 2h/l 5 1 switching from 18.5 to 238.8will result. Making this switching symmetrical in termsof tilt angle (q1 5 2q2) requires that the shift betweenthe top and the bottom electrodes be equal to one-half theelectrode period, d 5 0.5l. These refractive-index distri-butions are presented in Fig. 6 together with the distri-bution that is found when all the top electrodes have 1V0potential and all bottom electrodes, which are shifted byhalf of the period, have 2V0 potential. As we can see,switching to this state gives us an untilted grating with a

Fig. 6. Contour plots of the refractive-index distribution for (a)d 5 1.5l, (b) d 5 0.5l, and (c) when all top electrodes have1 V0 potential and all bottom electrodes have 2 V0 potential.

The calculation was made for LiNbO3 wafer (e11 5 85, e335 29), e 5 1, 2h 5 0.5l, V0 5 3V, h 5 1 mm; ne 5 2.3, r335 30 3 10 2 12 m/V, and a/l 5 0.5. The induced refractive in-dex varies as (a), (b) 2.29801 < n < 2.30199; (c) 2.29977 < n< 2.30192.

462 J. Opt. Soc. Am. B/Vol. 18, No. 4 /April 2001 Kulishov et al.periodicity that is two times less than the distributions inFigs. 6(a) and 6(b). To get the last plot we used the re-sult from Ref. 9. The results obtained demonstrate thatthe design under consideration provides a good possibilityof electronic switching (at least three states throughsimple voltage commutation) and tuning by adjustment ofthe magnitude of the effective index variation throughbias voltage V0 . If buffer layers (cladding) are madefrom different materials or have different thicknesses,there will be different modes for cladding-mode scatteringand absorption that can be controlled by switching of thecoupling direction.

The design described can be realized as an EO trans-missive (or long-period) tunable phase grating thatcouples light from the forward-propagating guided coremodes to the forward-propagating cladding modes andthe radiation field.1315 The overlap integral that gov-erns the interaction can be adjusted by electrode voltagecontrol (V0) or by voltage commutation switching. Fortransmission gratings, power is exchanged between thecladding and the core modes periodically, so the gratinglength governs the bandwidth of the coupling to the clad-ding modes. In our electro-optic grating concept thelength can easily be controlled, because coupling does notoccur without the presence of voltage at the electrodes.If the total number of electrodes is too large, we can al-ways adjust the number of active electrodes.

Another peculiarity of the transmissive gratings is thatthey are sensitive to differences in the propagation con-stants of the core and the cladding modes. Any EO in-duced change in the core index will result in a shift in thepropagation constants and thereby strongly affect theresonance wavelength. Therefore, for a typical fiber, thetransmissive grating is approximately 100 and to 1000times more sensitive than the reflective tilted grating tochanges in the propagation constants of the core and thecladding modes.16

One can use these properties, for example, to build anactive gain-equalizing filter for dynamic erbium-doped fi-ber amplifiers. In the various operating conditions ofwideband wavelength-division multiplexing systems, theuse of such filters is essential to equalize both the signalpower and the signal-to-noise ratio of multiple channelsover a wide spectral bandwidth.17

5. EO SIDE-TAP GRATINGSIf we achieve submicrometer periodicity of EO inducedgratings, a reflective type of tilted (or side-tap) gratingcan be created on the basis of our concept. This is a chal-lenging task; however, it is already technologicallyfeasible.7,8

It has been recognized that cladding and radiationmode coupling can be enhanced and to a certain extentcontrolled if a tilt is provided in the fringes of the phasegrating. Controlling a tilt angle allows us to controlwidth of the loss spectrum, the separation of the wave-length region at which maximum radiation mode couplingoccurs from the region at which Bragg reflection takesplace.13 Tilt affects reflection by effectively reducing thefringe visibility; however, exceeding 12 of fringe tilt isnot practical, because doing so will decrease the effectivegrating fringe visibility to zero.18 Important propertiesof titled gratings are their stability and intrinsic low sen-sitivity to temperature. The potential applications arenumerous, e.g., in-fiber noninvasive taps, spectrum ana-lyzers, and mode converters.

Figure 7 shows us the variations of the first (fundamen-tal) and the second spatial harmonics of the normal com-ponent of the electric field in the middle of the core (z5 0) as a function of the shift distance (d) between thetop and the bottom electrodes for a number of values ofthe electrode-distance-to-electrode-period ratio (2h/l).As we can see, the first harmonic achieves its maximumvalue for d/l 1.45 and its minimum value for d/l 0.5; however the difference between these extremes isnegligible for 2h/l > 1. At the same time, as the secondharmonic is at least an order of magnitude smaller thanthe first one and changes its sign to cross its zero value[Fig. 7(b)]. Another important dependence of the firstharmonic on electrode duty ratio is presented in Fig. 8,which tells us that electric field can be increased by;20%, increasing the electrode width from 0.35l to 0.7l;however, the electrical strength of the structure also hasto be taken into account.

The concept of a tilted electro-optic grating can also berealized in planar geometry, as is shown in Fig. 9. Thisplanar design can be more practical for certain applica-tions. Although the results of a two-dimensional distri-bution of the double-sided electrode structure describedabove cannot be applied for a quantitative description ofthe three-dimensional refractive-index distribution of theplanar geometry, qualitatively they can be used for theanalysis.

Fig. 7. (a) The first and (b) the second harmonic of the normalcomponent of the electric field in the middle of the core (z 5 0)as a function of the relative shift between the top and the bottomelectrodes for several values of the electrode-distance-to-electrode-period ratio: (2h/l 5 0.4, dotted curves; 2h/l 5 0.5,solid curves; 2h/l 5 0.75, dashed curves; and 2h/l 5 1 dotteddashed curves). The calculation was made for LiNbO3 wafer(e11 5 85, e33 5 29), e 5 1, a/l 5 0.5.

Kulishov et al. Vol. 18, No. 4 /April 2001 /J. Opt. Soc. Am. B 463An important issue for all designs for both long-periodand short-period gratings is guided-wave interaction withthe periodic set of conductive electrodes. We plan to ad-dress precisely this issue during an optical simulationthat will be the subject of a future paper. However, it isclear that an appropriate cladding thickness has to bechosen to minimize this interaction. There is a trade-offhere, because a thick cladding will result in a weak elec-tric field and a refractive-index change in the core. An-other solution for minimizing this interaction is replace-

Fig. 8. First harmonic of the normal component of the electricfield in the middle of the waveguide (z 5 0) as a function of elec-trode duty ratios: (2h/l 5 0.5, squares; 2h/l 5 0.75, diamonds;2h/l 5 1, crosses). The calculation was made for LiNbO3 wafer(e11 5 85, e33 5 29), e 5 1, d/l 5 1.5.

Fig. 9. Planar geometry of the tilted electro-optic grating: (a)top view, (b) cross section.

Fig. 10. Basic structure of our waveguide output coupler withan EO induced tilted grating: ITO, indium tin oxide.ment of metal electrodes by a heavily dopedsemiconductor materials such as indium tin oxide.

The achievement of high-efficiency coupling to andfrom integrated-optic waveguides is essential for exten-sive use of integrated optic circuits in industrial commu-nication and sensor systems. It is known that gratingcouplers made by the usual reactive-ion etching or ion-beam etching exhibit an efficiency that is too low to be ofpractical use, as only part of the light power can be effi-ciently coupled out into a desired diffraction order. Toobtain high coupling efficiency by using waveguide grat-ings one should maximize the percentage of total guided-wave power exiting the waveguide in the direction of thesuperstrate (i.e., the branching ratio).

Generally, methods of increasing the branching ratioinclude shaping the grating groove profile,19 incorporat-ing a highly reflective substrate,20 and fabricating corru-gations on both the upper and the lower surfaces of thewaveguide.21 In our design we can combine all threemethods to achieve the highest output. Instead of grooveshaping, the use of which is not always realistic becausethe requirements for a groove depth that will produce aspecific groove profile may not permit independent controlof the grating diffraction efficiency, a blazing effect can beobtained by the EO induced tilt. The electrodes from thesubstrate interface can be made from highly reflectedmetal (Ag or Gd), and electrodes from the superstrate sidecan be transparent (indium tin oxide). Lastly, the finalthickness of the reflective substrate electrodes results innatural corrugation on the upper and lower surfaces ofthe waveguide, as shown in Fig. 10. The proposed designallows us to adjust the outcoupling direction electroni-cally by controlling DV voltage (22V0 < DV < 0), whichgives us more flexibility to control the light than do theknown designs.22

6. CONCLUSIONSIn this paper a new concept of an electro-optically switch-able waveguide grating has been proposed. A rigorouselectrostatic analysis has been developed as a tool foranalyzing the refractive-index distribution induced bytwo shifted IDE structures. Optimization of structurecapacitance, which directly affects the time response, canbe made on the basis of the calculation results. From theanalysis it can be shown that the proposed design pro-vides an opportunity to switch EO the direction of the in-duced grating wave vector. It has been shown that onecan use the concept to build EO controllable transmissive(long-period) and reflective (short-period) tilted gratingsand couplers in both transverse and lateral configura-tions.

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