Finite-element analysis of lossyTE–TM modes in metal-clad optical waveguides

Christos Themistos, B. M. Azizur Rahman, and Kenneth Thomas Victor Grattan

Finite-element analysis employing the scalar and vector H-field formulations and with the aid of theperturbation technique is used to calculate the TE–TM complex propagation characteristics of integratedoptical devices in gallium arsenide, lithium niobate, and silica fiber, incorporating a lossy metal cladding.The propagation and attenuation properties of several types of metal-clad planar optical waveguide,which exhibit surface-plasmon properties for the TM polarization, are reviewed, and the modal losscaused by the metal cladding in a titanium-diffused lithium niobate electro-optic directional couplermodulator, an indium gallium arsenide phosphide-based TE–TM optical polarizer, and a submicronmetal-clad silica fiber suitable for near-field optical scanning microscopy is calculated. © 1998 OpticalSociety of America

OCIS codes: 230.0230, 230.7370, 250.7360, 260.5430.

1. Introduction

The characterization of optical waveguides, whichare the key elements in the design of integrated op-tical devices, requires the accurate determination ofthe impact of various material parameters and fab-rication tolerances, for example. During the earlyyears of the development of the technology, the esti-mation of loss and gain was not considered criticalsince it was maintained at low levels, owing to thesimplicity of the structures and the properties of thematerials. However, a loss and gain analysis hasbecome of considerably greater importance with theintroduction of new laser technology and integratedoptics design that has enabled the fabrication of com-plicated structures in which various metallic elementsand active regions are combined in a large-scale inte-gration of the structure.

Several optoelectronic devices as described re-cently, such as optical polarizers,1 electro-optic mod-ulators,2 scanning microscopes,3 and potentiallyhighly sensitive optical sensors, incorporate metalmaterials either as metal electrodes or as a metalcladding. Practical metallic elements are not per-

The authors are with the Department of Electrical, Electronic,and Information Engineering, City University, NorthamptonSquare, London EC1 0HB, United Kingdom.

Received 25 November 1997; revised manuscript received 30March 1998.

0003-6935y98y245747-08$15.00y0© 1998 Optical Society of America

fect conductors but suffer a small amount of loss.Therefore the modeling of the loss in the analysis ofoptical waveguides incorporating metal films and theinteraction of the metallic films with dielectric mate-rials, for accommodating guided optical waves, is con-sidered important for the accurate design of thevarious devices, which themselves have a range ofapplications. In optical waveguide analysis metallicelements can be represented by a complex dielectricconstant em with a negative real part. The study ofthe propagation and attenuation characteristics ofthe metal-clad optical waveguide,4 which is the basicelement in an integrated optical circuit, can be usefulin the characterization of more complicated metal-clad structures that exhibit properties based on thesame principles. Here initially the propagation andattenuation characteristics of several types of planarmetal-clad optical waveguide are studied to identifycertain of their properties, such as the existence ofsurface-plasmon modes,5 a knowledge of which is im-portant in the design of the various practical struc-tures.

The finite-element method ~FEM!,6 which is a pop-ular numerical approach for the solution of manyengineering problems, is currently recognized as apowerful tool for the analysis of several opticalwaveguide structures, particularly those with arbi-trary shapes, index profiles, nonlinearities, andanisotropies. The vector H-field variational formu-lation7,8 has been used widely in the solution of sev-eral loss-free optical waveguide problems. Recentlythe scalar and the vector H-field variational formu-

20 August 1998 y Vol. 37, No. 24 y APPLIED OPTICS 5747

a

matifi

en

t

of

5

lations of the FEM have been employed in conjunc-tion with the perturbation technique9,10 in thenalysis of planar and three-dimensional ~3-D! opti-

cal waveguides with small to medium loss–gain, suchas optical waveguides with surface-plasmon proper-ties11 and semiconductor laser structures.12 Herewe review the analysis of metal-clad optical struc-tures to investigate the propagation characteristics ofcomplex 3-D metal-clad structures with practical ap-plications in the optics industry.

2. Metal-Clad Planar Waveguides

The propagation characteristics and modal-field pro-files for different types of metal-clad planarwaveguide are first studied for TE- and TM-polarizedmodes, and particularly the surface-plasmonmodes,13,14 which are TM-polarized electromagneticwaves supported by a single or multiple metal-dielectric interface. Surface-plasmon mode proper-ties are used in a wide range of device applications,such as in optical polarizers or highly sensitive eva-nescent optical sensors.

First, a single metal-dielectric interface, which isthe most basic structure that can support a guidedwave, was examined. The waveguide consisted ofan aluminum ~Al! layer with a complex dielectric con-stant em attached to a dielectric layer with a refrac-tive index ng at an operating wavelength of l 5 1.3

m. With the problem solved by use of the scalarpproximation of the FEM with perturbation, onlyhe lowest TM mode, TM0, could propagate along thenterface. Figure 1 shows the normalized magnetic-eld profile ~Hx! in the direction transverse to the

propagation ~y! of the TM0 optical mode, the onlyguided mode, for different values of the refractiveindex of the dielectric material. In the metal regionthe field decays very rapidly, whereas in the dielectricregion the decay depends on the value of the refrac-

Fig. 1. Variation of the normalized field profile for the TM0 modef a single metal-dielectric interface with the transverse directionor several values of the refractive index of the dielectric.

748 APPLIED OPTICS y Vol. 37, No. 24 y 20 August 1998

tive index ng of the dielectric material. As the re-fractive index ng increases, the mode becomes moreconfined with a faster decay of the field in the dielec-tric region, while the rapid decay in the metal regionremains unchanged.

Next, two different planar metal-clad structuresthat support coupled surface-plasmon modes wereexamined: first, when a thin film was sandwichedbetween the two dielectric materials, and second,when metallic films were deposited on each side of anormal dielectric.

The first three-layer structure under considerationconsists of a thin metal film with a dielectric constantem 5 er 1 jei bounded at y 5 0 and y 5 t by twosemi-infinite lossless dielectric materials with refrac-tive indices n1 for y . t and n3 for y , 0. The firstxample considered is a symmetrical waveguide with1 5 n3 and em 5 2144 2 j28.8, representing a thin

Al film at wavelength l 5 1.3 mm. Figure 2 showsthe effective-index variations with the metal thick-ness ~t! and with inset modal-field profiles for differ-ent values of the cladding refractive index. Twobounded supermodes are examined, the first corre-sponding to the antisymmetrical ~odd! ~Ab! and thesecond to the symmetrical ~even! ~Sb! supermode.

Both the modes have a higher effective index ne ~ne5 byk0! than the common cladding refractive index~n 5 n1 5 n3! of the two ~top and bottom! claddingregions. Here b is the propagation constant and k0is the wave number. From the variation of the ef-fective index for n1 5 n3 5 1.452, shown by a solidcurve in Fig. 2, it can be seen that, as the film thick-ness increases, the antisymmetrical mode becomesless confined, since the effective index decreases andthe field spreads further into the cladding ~shown inthe second upper insert of Fig. 2!. In contrast withhe antisymmetrical mode, the effective index ~byk0!

of the symmetrical mode increases with the increaseof the metal thickness and the field becomes moreconfined, but showing a larger central dip as the two

Fig. 2. Variation of the effective indices for a symmetric surface-plasmon structure, in which a thin metal film is sandwiched be-tween two dielectrics.

gmde1a

wialosmtrr

mcamafp

adgiw

tt

metal-dielectric interfaces move apart ~shown in thesecond lower insert of Fig. 2!. The effective index ofthe symmetrical mode reduces and approaches thecladding refractive-index value as the metal thick-ness decreases and the mode becomes more weaklybounded. As the metal thickness t increases fur-ther, the two supermodes behave like two weaklycoupled surface modes, and the two effective indicestend to reach the effective index of the mode sup-ported by a single metal-dielectric interface. As the~identical! refractive index of both the cladding re-ions decrease to 1.447, the effective index of eachode shifts downward by an amount equal to that

ecrease, as shown by the two dashed curves. Theffective-index values for the above case ~n1 5 n3 5.447! agree well with those reported by Johnstone etl.15 and also with the results obtained by use of an

analytical approach,10 which are not shown here.The attenuation constants for such structures are

also calculated by use of the scalar approximation ofthe FEM with perturbation. In Fig. 3 the solidcurves represent the attenuation curves for the sym-metrical and antisymmetrical modes in the symmet-rical structure, which very much resemble theeffective-index curves shown earlier in Fig. 2, wheren1 5 n3 5 1.447. The attenuation constant de-creases monotonically for the antisymmetric-boundmode ~the first supermode! with the metal thickness,

hereas for the even mode ~the second supermode! itncreases. For a given value of the metal thickness

higher-power fraction is confined inside the metalayer for the first supermode, compared with the sec-nd supermode, so the modal loss is higher for thisupermode. For this reason the even or evenlikeodes are also known as long-range modes, whereas

he odd or the oddlike supermodes are termed short-ange modes. The attenuation constant valueseach that of the surface mode supported by a single

Fig. 3. Variation of the normalized modal loss coefficient versusthe metal thickness for the first two supermodes for different val-ues of the lower-cladding refractive index, in which a thin metalfilm is sandwiched between two dielectrics.

etal-dielectric interface when t is large. Dottedurves represent attenuation curves for the evenlikend oddlike modes when the structure is not sym-etrical, i.e., when the refractive indices of the upper

nd lower cladding are not identical. Modal lossesor the oddlike and evenlike modes are nearly inde-endent of the individual refractive-index values.Apart from the structure discussed above, in whichthin metal film was embedded between two normalielectrics, surface-plasmon waves can also propa-ate in waveguides with an opposite arrangement,.e., where a thin planar-dielectric medium is sand-iched between two metallic films.The structure under consideration is a symmetrical

hree-layer planar waveguide, where a normal dielec-ric film of refractive index ng 5 2.12285 is bounded

by two identical metal films with dielectric constantem 5 24 2 j0.5414 at an operating wavelength of l 50.6328 mm. First, the variation of the effective indexwith the normalized dielectric film thickness k0a wasexamined for the first two bounded supermodes, andresults are presented in Fig. 4. As can be seen fromthe graph, the supermode with the highest effectiveindex is an evenlike mode, whereas in Fig. 2 for theopposite structure the first supermode is oddlike.For a small dielectric film thickness 2a, the effectiveindex ne is very high, but as the film thickness in-creases the effective index ne reduces to the value ne5 2.1289. The second supermode in this type ofstructure is an oddlike mode, whereas in the oppositecase in Fig. 2 it is an evenlike mode. For a smallvalues of the dielectric film thickness 2a the mode isin the cutoff region, and, as the film thickness in-creases, the effective index increases to a value veryclose to that encountered for the first supermode,which in this case is ne 5 2.1252. For a large metalfilm thickness the effective indices of the oddlike andthe evenlike modes approach from above and below,respectively, the effective index of the single metal-

Fig. 4. Variation of the effective index with the dielectric filmthickness for a symmetrical surface-plasmon structure, in which aplanar-dielectric layer is sandwiched between two metal films.

20 August 1998 y Vol. 37, No. 24 y APPLIED OPTICS 5749

sgnlpfs

st

5

dielectric interface, which was calculated here as ne 52.12695.

Next, the attenuation characteristics of the firsttwo supermodes, shown in Fig. 5, are considered.The attenuation constants for both the two super-modes decrease as the normalized film thickness k0aincreases, unlike the dielectric-clad structure in Sec-tion 1, in which the attenuation constant was increas-ing for the evenlike and decreasing for the oddlikemodes. In this type of structure the thickness of themetal claddings, which are the lossy regions, does notchange, and therefore the attenuation constants de-pend mostly on the optical field intensity at themetal-dielectric interfaces. For the first supermode,the even supermode, the attenuation constant islower than it is for the second, the odd supermode.As the dielectric film thickness increases, the twonormalized attenuation constants converge to a cer-tain value that corresponds to the attenuation con-stant of a single metal-dielectric interface, as in thepropagation characteristics, and this was calculatedto be ayk0 5 0.16281. This can be explained, again,from the field profiles of the two supermodes, whichare decoupled to two independent surface-plasmonmodes that propagate at the two interfaces for largedielectric film thickness.

The FEM, in conjunction with the perturbationtechnique, was applied in several types of planarwaveguide involving surface-plasmon modes, and itproved to be an accurate approach in dealing withlow-loss devices for the determination of their gain–loss properties. The accuracy of the above approachcan be quite good for gain–loss ranges to as high as10000 dBycm.10 Surface-plasmon modes also existin a wide range of device applications with metalcladdings, in which two-dimensional modal analysisis required to find the modal interaction betweensuch modes and optical guided modes, such as foroptical polarizers, modulators, and sensors. The

Fig. 5. Attenuation characteristics of a symmetrical metal-cladsurface-plasmon structure with dielectric material sandwiched be-tween two metal films.

750 APPLIED OPTICS y Vol. 37, No. 24 y 20 August 1998

complex propagation characteristics of some practical3-D metal-clad waveguide structures are examined inSections 3 through 5.

3. TE–TM Polarization Mode Splitters

TE–TM polarization mode splitters are required inmany systems used in optical communications to sep-arate effectively the TE and TM polarization of theoptical wave. A cross section of a TE–TM modesplitter based on two parallel rib waveguides, with ametal cladding on one of the ribs, is shown as aninsert in Fig. 6. When a composite TE–TM opticalwave is launched at the input of the nonmetal rib,there is a power transfer to the metalized rib, owingto the coupling. Since the two rib regions in themode-splitter structure, apart from the presence of ametal cladding on one of them, have identical refrac-tive indices, in the case of the TE mode the device canoperate as a synchronous directional coupler, becausethe isolated mode has approximately the same prop-agation constant in each guide core region. There-fore, with an adjustment in the length of the device,the TE mode launched at the nonmetal rib can befully coupled to the metallized rib waveguide. Onthe other hand, for the TM polarization the twoguides are not identical owing to the presence of gold~Au! cladding on one of them. The propagation con-tants for the TM-polarized light in the two isolateduides are unequal. The TM mode launched in theonmetal rib is not fully coupled to the lossy metal-

ized rib because of the phase mismatching, so itropagates mostly along the nonmetal rib, and there-ore the two modes are decoupled at the output of thetructure. Albrecht et al.16 demonstrated that such

a structure can operate in two distinct modes depend-ing on the degree of lateral confinement of the opticalrib waveguide before metallization. They showedthat, for strong confinement, metallization can gen-

Fig. 6. Variation of the effective index with the increase of the ribheight ~d! for the TE and TM modes of the polarization modeplitter ~shown as an insert! for different separations ~s! betweenhe two ribs.

ctaI

jWslwvs

ad

dr

filmctorspT

mcim

tsdamrct

xr

erate an asymmetrical coupler and that, for weakconfinement, the metallized waveguide can be madeto operate as guiding for TE but nonguiding for TMpolarization.

The mode polarization splitter examined ~with itsross section shown as an insert in Fig. 6! consists ofwo parallel rib waveguides, with an indium galliumrsenide phosphide ~InGaAsP! active region and annP substrate with refractive indices ng 5 3.38 and ns

5 3.17, respectively, and an Au metal cladding on topof that with a complex refractive index nm 5 0.18 210.2, at an operating wavelength of l 5 1.55 mm.

e performed a modal analysis of the above crossection, using the FEM with perturbation, to calcu-ate the propagation and attenuation characteristics,hich can be applied in the determination of thearious design parameters of the mode polarizationplitter.Next, the properties of the fundamental TE mode

nd the two low-order TM modes were examined forifferent values of the separation s between the two

ribs. From the propagation characteristics shown inFig. 6, it can be seen that the TEeven, or TE00 mode~where TExy denotes the zero crossings along the xand y axes! has the highest effective index, which

oes not change significantly with the increase of theib height d but decreases slightly when the separa-

tion is increased. As the separation between the tworibs ~s! increases, the effective index of the TMevendecreases while that of the TModd increases. Theeffective index of the TMeven or TM01 mode is higherthan that of the TModd, or TM11 mode, and both in-dices increase with the increase of the rib height d.Note that there also exists a TM mode with a highereffective index than those presented, but this was notconsidered here because it is a pure surface-plasmonmode located at the metal-dielectric interface and itdoes not interact with the guided dielectric mode ofthe active region, and therefore it is unsuitable foroptical polarizing applications.

Figure 7 shows the Hx-field profile of the TM01mode along the x axis in the center of the InGaAsPregion for different values of the rib height for a sep-aration s 5 0.7 mm. As the rib height increases, the

eld intensity at the side of the metal rib, which isower than that of the nonmetal, increases, since the

etal cladding has moved away from the dielectricore region of the guide. On the other hand, whenhe rib height increases, the field intensity at the sidef the nonmetal rib decreases, with the tendency toeach the height of the field intensity at the otheride, as the phase matching between the guides im-roves. Therefore at a large rib height the coupledM01 mode tends to approach the shape of the TE00

mode, which is shown as an insert for rib heights d 50.6 mm ~solid cure! and d 5 1.0 mm ~dotted curve!. Itcan be seen that for the even TE00 supermode thefield intensity is approximately equal below each riband varies by a negligible amount with the variationof the rib height d. In the case of the TE00 mode, inwhich the two coupled modes at the two rib sidespropagate with an almost identical propagation con-

stant, the device operates like a synchronous direc-tional coupler.

In the design of the mode polarization splitter theoptimum device length should be an integer odd oreven multiple of the coupling length for the TE andthe TM modes. One can achieve this by finding theoptimum separation s between the two ribs and the

etal-clad rib height d, but also by taking into ac-ount the modal loss of the structure, which is anmportant factor in the overall power transfer perfor-

ance of the device.The variation of the attenuation characteristics of

he TE and TM modes, for the polarization modeplitter with the increase of the rib height and for twoifferent values of the separation s, was considerednd is presented in Fig. 8. At a low rib height theodal loss for the TMeven mode increases until it

eaches an absorption peak, and then it starts de-reasing rapidly. For a smaller rib height, wherehe metal layer is nearer the guide core, the TModd

mode suffers more attenuation because the opticalpower is more confined in the metal-clad rib. TheTModd mode, which is more confined in the metal-cladwaveguide side, has a higher attenuation constantthan the TMeven mode and decreases monotonicallyas the rib height increases, not showing an attenua-tion peak over the range examined. For a large ribheight d the effect of the metal cladding is less, andtherefore the modal loss is low, and the two isolatedguides are nearly phase matched. The TE mode hasthe lowest attenuation constant, it being approxi-mately 1000 times lower than that of the TMevenmode, and the attenuation constant decreases as therib height increases in a similar way to that of the TMmodes.

Fig. 7. Hx-field profile for the evenlike TM01 supermode along theaxis in the center of the active region for different values of the

ib height at a separation s 5 0.7 mm. ~The TEeven mode profile isshown as an insert.!

20 August 1998 y Vol. 37, No. 24 y APPLIED OPTICS 5751

tgt

csmltc

5

4. Metal-Clad Fibers for Optical Scanning Microscopy

Single-mode optical fibers are widely used in long-distance telecommunications and are well developed,providing low-loss and high-bandwidth transmissionmedia. However, metal-clad optical fibers with sub-micron diameter are currently being used in near-field scanning optical microscopy,17 a technique inwhich light is transmitted through a metal-coatedtapered fiber with a submicron aperture at the endused as the light-emitting probe. A subwavelength-sized spot is formed on an opaque screen, which isscanned over an object to generate a super-resolutionimage. The resolution is closer to that of a scanningelectron microscope than a conventional microscope,and it has many advantages since it can operate inair, is not limited to conductive materials, and pro-vides information on optical properties of the samplerather than electrical ones.

Here the propagation of the fundamental guidedoptical TM- and TE-polarized modes suitable for op-tical scanning microscopy in metal-clad circularwaveguides is examined. Complex propagationproperties and field profiles are investigated, with aview to a better understanding of the propagationproperties of these important fibers.

A metal-clad optical fiber is considered, with a coreand a cladding complex dielectric constant given byeco 5 2.16 and ecl 5 234.5 2 j8.5, respectively, at anoperating wavelength of l 5 0.488 mm, in which thecross section is shown as an insert in Fig. 9. Figure9 also shows the variation of the effective index ne ofhe fundamental and the higher-order symmetrical,uided optical quasi-TE and quasi-TM modes, withhe core radius rco, where the axial variations of the

field profiles are shown as inserts. As the core ra-dius rco, increases, the effective index of the TM00mode decreases, and its limiting value for a very largeradius ~rco3 `! can be approximated to the value ofthe effective index of a two-layer planar waveguide

Fig. 8. Variation of the attenuation constant for the TM and TEmodes of a polarization mode splitter with the increase of the ribheight for different values of the separation s.

752 APPLIED OPTICS y Vol. 37, No. 24 y 20 August 1998

npl consisting of a single metal-dielectric interface.It was shown earlier that a surface-plasmon modecan be supported by a single metal-dielectric inter-face, and its effective index is also higher than therefractive index of the dielectric layer. The effectiveindex of such a planar waveguide npl with the refrac-tive indices of the dielectric-metal layers, which aresimilar to those of the core–radius regions of themetal-clad fiber, was calculated to be npl 5 1.51853.It is also well known that the field is maximum at themetal-dielectric interface and decays exponentially inboth the boundary regions. The metal-clad fiber canbe considered as a folded-back metal-dielectric inter-face. The field in the outer metal layer decaysquickly, whereas the slowly decaying field in the di-electric region couples to a similar decaying field atthe diametrically opposite side to form a dip at thecenter. The effective index for the TM00 mode in-reases with the reduction of the radius in a wayimilar to the coupled surface-plasmon even super-ode, which can be supported by a thin dielectric

ayer sandwiched between two dielectric layers. Ashe core radius increases, the influence of the metalladding becomes negligible for the TM01 optical

modes and its effective index tends to reach the re-fractive index of the core nco ~nco 5 1.4697!. As thecore radius decreases, the effective index of the abovemode decreases and approaches the cutoff region.

The effective indices of the TE00 and TE01 decreaseas the core radius is reduced, and all the modes reacha cutoff region after a certain value. It is well knownthat the TE modes are not significantly affected bythe metal layers and their modal properties are sim-ilar to those of other TE modes in dielectric opticalwaveguides. The fundamental TE00 optical mode

Fig. 9. Variation of the effective index with the core radius for thefundamental TE and TM axisymmetric modes of a metal-clad op-tical fiber.

d

ctc

f

iigao

rst

Ac

has a larger effective index and reaches cutoff at asmall core radius, compared with the higher-orderTE01 mode. The TE00 has characteristics that re-semble those in a standard optical fiber. The fieldprofile is similar to a well-confined fundamentalmode in a dielectric waveguide with negligible field inthe cladding region. There is a high field intensityat the center of the dielectric core, and the opticalfield decreases monotonically along the radial direc-tion without any sharp field variations at the metal-dielectric interface. On the other hand, unlike theTM00 mode, which has a more uniform field profile inthe core region with a sharper field decay in the clad-ding, the TE00 does not show a similar feature. Ad-

itionally, the higher-order TE01 optical mode showssome similarities to the TM01 mode illustrated, sinceboth the modes have a high field intensity in thecenter of the metal-clad optical fiber and an appre-ciable field at the metal-dielectric boundaries alongthe circumference. The main difference betweenthem is the field intensity at the dielectric-metal in-terface. For the TM01 mode there is a sharp changeof the field intensity at the interface, but for the TE01optical mode that change is rather smooth.

The above optical modes examined are all symmet-ric and used in near-field scanning optical micros-copy, depending on the optical field requirements,i.e., whether a strong optical field is required in thecenter of the probe or at the rim of its aperture.However, these modes also suffer from attenuation,and the normalized attenuation curves for the fun-damental and the higher-order symmetric TE- andTM-polarized modes are shown in Fig. 10. TheTM00 has the highest normalized attenuation con-stant for a small radius of the metal-clad opticalwaveguide, because at this range the optical field ishigher at the dielectric-metal interface, and thereforethere is a greater amount of the optical field confinedin the lossy material, which is the metal cladding.The total power confinement in the fiber core is alsosmaller as the core dimension is reduced. All thenormalized attenuation curves decrease as the coreradius increases, since the area of the dielectric corebecomes larger than that of the metal-clad area withappreciable field and the optical field in the latter is

Fig. 10. Attenuation characteristics for the symmetric TE andTM optical modes of a metal-clad optical fiber.

reducing. The TM01 optical mode has lower attenu-ation characteristics because, as discussed earlier,the field is more confined in the lossless dielectric corearea.

The normalized attenuation constant ~ayk0! de-creases for all the TE-polarized modes at a similarrate as the core radius increases. The fundamentalTE00 optical mode has the lowest attenuation char-acteristics, since there is no appreciable field inten-sity in the lossy metal cladding. However, the TE01mode has a higher normalized attenuation constant,since there is a finite field intensity at the dielectric-metal interface and inside the metal cladding. Notealso that the fundamental TE-polarized TE00 modesuffers much lower attenuation than the fundamen-tal TM-polarized TM00 mode.

5. Electro-Optic Directional Coupler Modulator

Directional couplers, which are the basis of severalguided-wave devices, mainly used for optical switch-ing networks, can also be used as intensity modula-tors. An example of a Db coupler modulator is alsogiven here, operating on the principle that the ap-plied modulating field changes the refractive index inthe two guides, such that the change is antisymmet-ric, and this then affects the lightwave propagation inthe two guides ~hence the name Db modulator!, theoupling length, and the phase matching betweenhem, which also affects the power-coupling effi-iency.

The development of an accurate numerical modelor optimizing a titanium:lithium niobate ~Ti:

LiNbO3! directional coupler structure requires theconsideration of several fabrication parameters.The effect of the lossy metal electrodes on the opticalproperties of the above structure for Al and Au ma-terials is examined here. The metal electrode de-sign is an important issue in maximizing the overlapbetween the optical and electric fields, which one canoptimize by varying certain parameters, such as theelectrode placement and the buffer layer thickness.

A Ti-diffused directional coupler modulator with aguide width w of 9 mm, where the separation betweenthe guides s is 7.5 mm and the electrode separation hs 5 mm, and which is shown as an insert in Fig. 11,s considered, where y-axis symmetry is assumed forreater accuracy. The silica ~SiO2! buffer layer hasthickness d that is 0.2 mm. The extraordinary and

rdinary refractive indices for LiNbO3 are taken to bene 5 2.14 and no 5 2.16 at an operating wavelengthof l 5 1.56 mm, and the maximum change in theefractive index caused by the Ti indiffussion is con-idered to be 0.01 and 0.005 for the extraordinary andhe ordinary indices, respectively.

The modal loss of the TM-polarized mode for Al anu metal electrodes of thickness t 5 0.15 mm, with

omplex refractive indices nm 5 1.44 2 j16 and nm 51.55 2 j11.5, respectively, with the variation of thebuffer layer thickness d, was investigated. Fromthe attenuation curves shown in Fig. 11 it can be seenthat the modal loss decreases rapidly at first as thebuffer layer thickness d increases, but gradually its

20 August 1998 y Vol. 37, No. 24 y APPLIED OPTICS 5753

5

slope reduces and tends to reach a steady value at abuffer layer thickness larger than 0.17 mm, which canbe considered ideal for the modulator design. The Alelectrode always showed a higher attenuation thanthe Au electrode for the whole range examined, whichis to be expected. The modal loss for the TE-polarized mode, which is not shown here, is muchlower than that of the TM mode, as in all the otherstructures examined earlier, and therefore the effecton the optical properties is not considered critical.

6. Conclusions

The complex propagation characteristics for metal-clad planar optical waveguides and metal-clad opticalwaveguides with two-dimensional mode confinementwith practical applications have been presented.The surface-plasmon properties of planar metal-cladoptical waveguides were first examined, and lossanalysis of such structures by use of the scalar FEMwith perturbation was performed. The vectorH-field FEM formulation was then used for the char-acterization and the perturbation technique to accessthe modal loss of TE–TM polarization mode splittersin InGaAsP and metal-clad SiO2 fibers suitable fornear-field scanning optical microscopy, where the ex-istence of surface-plasmon modes was also recorded.Finally, the above approach was used to study themodal loss caused by the metal electrodes in a Ti:LiNbO3 directional coupler modulator. The FEM

Fig. 11. Attenuation characteristics for a directional coupler mod-ulator with the variation of the SiO2 buffer thickness d for Al andAu metal electrodes.

754 APPLIED OPTICS y Vol. 37, No. 24 y 20 August 1998

with perturbation has been shown to be a powerfuland efficient approach for the analysis of severaltypes of optical waveguide structure with arbitraryshape, material-type, and refractive-index distribu-tion.

References1. L. Sun and G. L. Yip, “Analysis of metal-clad optical waveguide

polarizers by the vector beam propagation method,” Appl. Opt.33, 1047–1050 ~1994!.

2. E. L. Wooten and W. S. C. Chang, “Test structures for charac-terization of electrooptic waveguide modulators in lithium nio-bate,” IEEE J. Quantum Electron. 29, 161–170 ~1993!.

3. L. Novotny, D. W. Pohl, and P. Regli, “Light propagationthrough nanometer-sized structures: the two-dimensionalaperture scanning near-field optical microscope,” J. Opt. Soc.Am. A 11, 1768–1779 ~1994!.

4. A. Reisinger, “Characteristics of optical guided modes in lossywaveguides,” Appl. Opt. 12, 1015–1025 ~1973!.

5. M. N. Zervas, “Surface plasmon-polariton waves guided bythin metal films,” Opt. Lett. 16, 720–722 ~1991!.

6. P. P. Silvester and R. L. Ferrari, “Finite elements for electricalengineers,” ~Cambridge U. Press, Cambridge, UK, 1991!.

7. B. M. A. Rahman and J. B. Davies, “Finite-element solution ofintegrated optical waveguides,” J. Lightwave Technol. LT-2,682–688 ~1984!.

8. B. M. A. Rahman and J. B. Davies, “Vector-H finite elementsolution of GaAsyGaAlAs rib waveguides,” IEE Proc.-Optoelectron. 132, 349–353 ~1985!.

9. C. Themistos, B. M. A. Rahman, and K. T. V. Grattan, “Finiteelement analysis for lossy waveguides by using perturbationtechnique,” IEEE Photon. Technol. Lett. 6, 537–539 ~1994!.

10. C. Themistos, A. Hadjicharalambous, B. M. A. Rahman, andK. T. V. Grattan, “Lossygain characterization of opticalwaveguides,” J. Lightwave Technol. 13, 1760–1765 ~1995!.

11. C. Themistos, B. M. A. Rahman, and K. T. V. Grattan, “Finite-element analysis of surface-plasmon modes for lossy opticalwaveguides by the use of perturbation techniques,” Appl. Opt.34, 7695–7701 ~1995!.

12. C. Themistos, A. Hadjicharalambous, B. M. A. Rahman,K. T. V. Grattan, and F. A. Fernandez, “Gainyloss characteri-sation of optical waveguide and semiconductor laser struc-tures,” IEE Proc.-Optoelectron. 145, 93–98 ~1998!.

13. A. D. Boardman, Electromagnetic Surface Modes ~Wiley, NewYork, 1982!.

14. A. Yariv and P. Yeh, Optical Waves in Crystals ~Wiley, NewYork, 1984!.

15. W. Johnstone, G. Steward, B. Culshaw, and T. Hart, “Fibre-optic polarizers and polarizing couplers,” Electron. Lett. 24,866–868 ~1988!.

16. P. Albrecht, M. Hamacher, H. Heidrich, D. Hoffman, H.-P.Nolting, and C. M. Weinert, “TEyTM mode splitters on In-GaAsPyInP,” IEEE Photon. Technol. Lett. 2, 114–115 ~1990!.

17. U. Durig, D. W. Pohl, and F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 ~1986!.