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abrication and comprehensive modeling ofon-exchanged Bragg optical add–drop multiplexers

ose M. Castro, David F. Geraghty, Brian R. West, and Seppo Honkanen

Optical add–drop multiplexers �OADMs� based on asymmetric Y branches and tilted gratings offerexcellent-performance in wavelength-division multiplexed systems. To simplify waveguide fabrication,ion-exchange techniques appear to be an important option in photosensitive glasses. Optimum OADMperformance depends on how accurately the waveguide fabrication process and tilted Bragg gratingoperation are understood and modeled. Results from fabrication and comprehensive modeling arecompared for ion-exchange processes that use different angles of the tilted grating. The transmissionand reflection spectra for the fabricated and simulated OADMs show excellent agreement. The OADM’sperformance is evaluated in terms of the measured characteristics of the Y branches and tilted gratings.© 2004 Optical Society of America

OCIS codes: 130.3120, 130.1750, 230.1480.

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. Introduction

everal filter technologies have been utilized to per-orm an add–drop operation optically in wavelength-ivision multiplexed systems. In recent years,esigns that combine the operation of Y branches asode splitters and tilted gratings as mode converters

ave been both demonstrated1–3 and modeled.4,5

mplementation of these optical add–drop multiplex-rs �OADMs� by ion-exchange techniques in photo-ensitive glasses has the potential of producing high-erformance operation in a compact device withimple fabrication procedures. The performance ofhese devices can be evaluated in terms of the opticalignal-to-noise ratio �OSNR� at drop and outputorts. The OSNR, in turn, depends on the spectralransmission �or reflection� characteristics of theilted grating and on the coupling parameters at the

branch.4,5

The operation of a Bragg reflective OADM requiresflat and transparent transmission spectrum �0-dB

oss� with a sharp dip at the wavelength to be

J. M. Castro �jmcastro@email.arizona.edu� and D. F. Geraghtyre with the Department of Electrical and Computer Engineering,he University of Arizona, 1230 East Speedway Avenue, Tucson,rizona 85721-0104. B. R. West and S. Honkanen are with Op-

ical Sciences Center, The University of Arizona, 1630 East Uni-ersity Boulevard, Tucson, Arizona 85721-0094.Received 30 April 2004; revised manuscript received 23 August

004; accepted 25 August 2004.0003-6935�04�336166-08$15.00�0

r© 2004 Optical Society of America

166 APPLIED OPTICS � Vol. 43, No. 33 � 20 November 2004

ropped. This type of filter response can be obtainedith a Bragg grating written normal to theaveguide. In that scheme, however, a circulator is

equired for physical separation of the dropped chan-el. A tilted Bragg grating written in the waist inhich two single-mode asymmetric branches con-erge �Fig. 1� does not require additional componentso separate incoming signals from the dropped signal.

The operation of the asymmetric Y branch with ailted Bragg grating has been explained in detaillsewhere.4–6 In an ideal asymmetric Y branch,hannels from the narrower branch excite only thedd modes of the waist. Tilted gratings in the waistperate at three different Bragg conditions and canroduce good coupling between odd–odd, odd–even,nd even–even modes. However, there is an anglet which the tilted grating maximizes the desireddd–even mode conversion while it minimizes thedd–odd coupling.4,5 At this angle, only the channelt the odd–even Bragg condition is reflected andropped. This channel, traveling backward as anven mode, ideally will couple only to the widerranch. Using the second Y-branch, one can add onehannel from the wider branch as it is coupled to theven mode of the waist and reflected with mode con-ersion to the narrow branch.Several designs to improve the performance of the

-branch Bragg OADM have been proposed. Accu-ate modeling of the fabrication process is necessaryor effective implementation of these designs. To ob-ain the transmission characteristics of an OADM

equires both grating and waveguide parameters.

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No direct comparison of designs and experimentsas been made for these Y-branch Bragg OADMs.aking such a comparison is particularly difficultith Y-branch Bragg OADMs that use ion-exchangedaveguides, because accurate modeling of ion-

xchanged waveguide index profiles is difficult.nly recently have advanced models that predict the

on concentration profiles with high enough accuracyeen developed.7 In the research reported in thisaper, we apply our advanced model for ion-xchanged waveguides to design Y-branch BraggADMs. The modeling results are compared witheasured data obtained from samples fabricatedith buried ion-exchanged waveguides and photo-ritten gratings. We demonstrate excellent agree-ent between models and experiments. The

ossibility of predicting the performance of theseADMs accurately offers great promise for taking

ull advantage of the attractive features of buriedon-exchanged waveguides in photosensitive glasses.

. Waveguide Fabrication

he samples were fabricated in the 2-mm-thick bo-osilicate glass BGG31 �n � 1.4574 at 1.55 �m� byg�–Na� ion-exchange techniques. Two sets ofamples were fabricated, as shown in Fig. 2. Set 1onsists of four double-sided asymmetric Y branches

ig. 1. Optical add–drop multiplexer: I, input Y branch; II,aist with tilted Bragg grating; III, output Y branch.

ig. 2. Samples in set 1 �double-sided Y branch with tilted grat-ng� and set 2 �single-sided Y branch�. Arbitrary scale and posi-

iion.

2

hat converge into a two-mode waist. There are tworanches as shown in Fig. 1, section I: the narrowernd the wider branches and the waist in the mask are, 3, and 5 �m, respectively. Set 2 consists of twone-sided, nontilted Y-branch waveguides with di-ensions similar to those of set 1. In the waist of

he OADM of set 1 a Bragg grating was photowritten.n those waveguides the waist direction was tilted bysmall angle with reference to the grating normal.he angle of tilt from the mask was 1°–4°.The first step in the fabrication of the waveguidesas to coat the glass with a 100-nm-thick Ti mask,hich was then patterned with the OADM design.ubsequently the mask was oxidized for 1 h in aodium nitrate salt melt at 380 °C and then placednto a 1:1 AgNO3:NaNO3 melt at 280 °C for 1 hour.ext, the Ti mask was removed and a field-assistedurial of the waveguides was performed for 5 min at75 V. Finally, the sample was cut and the endacets were polished.

The Bragg grating was photowritten by exposure ofhe glass for 12 min to 85 mJ of energy per pulse at48 nm at a repetition rate of 50 Hz. The gratingas 8 mm long by 2 mm wide; the grating provideddequate coverage for the waveguides in set 1. Dur-ng photowriting, an angular alignment error �equalor all waveguides� estimated at �0.5° was produced.he period of the grating was 535 nm ��g�, and thestimated index modulation ��n� was 3.7 104.

. Ion Exchange and Waveguide Modeling

abrication of buried ion-exchanged waveguides con-ists of two steps: thermal diffusion and field-ssisted burial. During diffusion, the exchange ofons of different sizes and polarizabilities produces ahange in the refractive index near the surface of thelass. Both processes are described by the equa-ion8,9

�C�t

�DA

1 � �C � 2C ��� C�2

1 � �C�

eEkT

C� , (1)

here C is the normalized concentration of the in-oming dopant, DA is its self-diffusion coefficient, e ishe electron charge, k is Boltzmann’s constant, T ishe absolute temperature, E is the applied externallectrical field, and � is defined as

� � 1 �DA

DB, (2)

here DB is the self-diffusion coefficient of the out-oing ion. Both self-diffusion coefficients DB and DAepend on the composition and the temperature ofhe glass. An important property of our ion-xchange model is that it takes into account the effectf nonhomogeneous conductivity that results in aonconstant electric field profile during the field-ssisted burial step.7 This effect is due to the differ-nt mobilities of the two ions, A and B �the incoming

on, silver, and the outgoing ion, sodium, in our case�.

0 November 2004 � Vol. 43, No. 33 � APPLIED OPTICS 6167

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The local change in refractive-index profile is pro-ortional to the concentration of ion A:

n� x, y, �� � nsub��� � dnmax���CA� x, y�, (3)

here nsub��� is the substrate index before ion ex-hange and dnmax��� is the change in refractive indexhat results when CA � 1, determined experimen-ally.

Electric field �m of each quasi-TE and -TM modeupported by the waveguide is found by solution ofhe Helmholtz equation

� 2 � k2��m � �m2�m, (4)

here k � k0n�x, y� � 2�n�x, y��� is the wave num-er and �m is the propagation constant of the mthode. We employ a semivectorial finite-differenceethod described in Ref. 10 to solve Eq. �4� with the

ppropriate discontinuity conditions of the normaleld components.

. Mode Profiles from Measurements and Modeling

rom the model, Eqs. �1�–�3�, and the fabrication pa-ameters described in Section 3, the index profile ofhe waist was obtained as shown in Fig. 3. Usinghis index distribution, we obtained the mode profilesnd effective indices of the even and odd modes of theADM waist from a numerical solution of Eq. �4�, asiven in Table 1 and shown in Fig. 4.To measure the intensity mode profiles of set 2

amples we used an input source of 1.55 �m and an

ig. 3. Index profile after diffusion and burial. Contours at 0.1,.3, . . . , 0.9 of dnmax.

Table 1. Measured Chara

Variable Unit Source

� deg Measured 1�oo nm Measured 1560.97�oe nm Measured 1563.43�ee nm Measured 1565.98neff,0 �ee��2�z� 1.46neff,1 �oo��2�z� 1.45Transmission dip �oo� dB Measured 16.8Transmission dip �oe� dB Measured 14.7

Transmission dip �ee� dB Measured 31.7

168 APPLIED OPTICS � Vol. 43, No. 33 � 20 November 2004

R camera with 256 gray-scale levels and 50 nm 50m pixel resolution. The even-mode profile was ob-ained when the power was launched in the widerranch. We used the narrower branch as input tobtain the intensity profile of the odd mode.Figure 5 shows the measured even and odd inten-

ity mode profiles. These small variations comparedith the modeled profiles were attributed to temper-ture changes during fabrication or to small errors inithographic precision.

. Grating Modeling

wo-mode waveguides with a tilted grating can havehree reflections: odd–even, odd–odd, and even–ven. Each reflection is centered at a differentavelength and has a different strength and band-idth. Although only the odd–even �or even–odd�

eflection is required for performing the add–dropunction, the strength of the other reflections that doot provide mode conversion can be reduced only forpecific angles. The design of Bragg gratings is ex-lained in great detail elsewhere.11–13 In the case oftilted grating in a two-mode waveguide, mode con-

ersion will occur at wavelength �oe if the period inhe propagation direction, �z, is given by

�oe � �z�neff�0� � neff�1��. (5)

he other wavelengths that satisfy the non-mode-onversion Bragg condition are

�oo � �oe

2neff�1�

�neff�0� � neff�1��, (6)

�ee � �oe

2neff�0�

�neff�0� � neff�1��. (7)

ig. 4. Modeled intensity profiles: �a� even mode, �b� odd mode.

tics of Add–Drop Devices

Value Precision

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he maximum reflection �transmission dip� at thisavelength is given by

Rab � �rab�2 � tanh2��abL�, (8)

here � is the coupling coefficient, calculated as

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��n�ab���. (9)

n Eq. �9�, � and �n are the free-space wavelengthnd the index modulation, respectively. The sub-cripts a and b represent odd and even modes, re-pectively, in the two-mode waveguide with the tiltedrating, � is the angle of the grating, and �ab��� is the

ig. 5. Measured intensity profiles: �a� even mode, �b� odd mode.

verlap integral13: fore the odd–even overlap increases. For an angle

norptquently the reflection of noise. It can be seen from

Fo

he strength and the bandwidth of the three reflec-ions depend on the index modulation, the gratingength, and the overlap integral. However, there isnly one factor that makes one property strongerhan the other two: the overlap integral betweenodes in the tilted plane of the grating. The values

f the overlap depend on the angle of the tilted grat-ng, the index profile, the index difference, theaveguide width, and the grating period and width.o perform the add–drop function properly, onehould tailor these variables to obtain maximumdd–even overlap integral while the odd–odd andven–even integrals are minimized.Using Eq. �10�, we obtained the values of the over-

ap integral for various angles of the tilted grating, asan be seen from Figs. 6 and 7 for the modeled processnd the fabricated samples, respectively. In the lat-er case, the amplitude mode profiles were calculatedrom the measured power ���a�x, y��2� of the modes.he even modes are symmetric, with constant phasecross the mode profile. For the odd modes �anti-ymmetric� a � phase shift occurs laterally across theiddle of the mode profile �Fig. 5�b��. Figures 6 andshow that the odd–even overlap integral at 0° is

ero because of the orthogonality of the modes, i

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hereas the overlap between similar modes �odd–dd and even–even� is unity. The increase in therating angle produces coupling between orthogonalodes owing to asymmetric perturbation. There-

ig. 6. Odd–odd �dashed curve�, even–even �dotted curve�, anddd–even �solid curve� overlap integrals as a function of the grat-ng angle from modeled mode profiles.

ear 2° the odd–odd overlap goes to zero and thedd–even overlap attains its maximum value. As aesult, at this angle it is possible to obtain optimumerformance for the OADM operation. Increasinghe angle increases the odd–odd overlap and conse-

ig. 7. Odd–odd �dashed curve�, even–even �dotted curve�, anddd–even �solid curve� overlap integrals as a function of the grat-

�ab��� ��� �a*� x, y�exp�i2�x tan�����z��b� x, y�dxdy

��� �a*� x, y��a� x, y�dxdy �� �b*� x, y��b� x, y�dxdy�1�2 . (10)

ng angle from measured mode profiles.

0 November 2004 � Vol. 43, No. 33 � APPLIED OPTICS 6169

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igs. 6 and 7 that the even–even overlap is biggerhan the odd–even overlap at the optimum angle;his will cause a large amount of even–even reflec-ion. Nevertheless, the reflected power can be neg-igible because of mode rejection of the Y branch.4–6

n practice, however, fabrication defects in the Yranch can lead to undesired coupling. The evenode traveling from the input port �narrow branch�

an couple to odd and even modes �Cneo, Cnee� of theaist. In addition, the even mode from the add port

wide branch� can couple to odd and even modesCweo, Cwee� of the waist. From the waist, the evennd odd modes can couple to even modes of the nar-ow and wide branches �Cnee, Cwee, Cnoe, Cwoe�.he combined effect of the Y branch and the tiltedrating in the OADM optical signal-to-noise ratio cane expressed as

OSNRdrop � 10 log� CneoRoeCwee

CneoRooCwoe � CneeReeCwee� ,

(11)

OSNRoutput � 10 log� CweeRoe

Cneo�1 � Roe�� . (12)

t can be seen from Eqs. �11� and �12� that to optimizehe OADM it is necessary to minimize the mode crossalk �Cnee, Cweo� and the odd–odd reflection �Roo�hile maximizing Cneo, Cwee, and the reflection withode conversation �Roe�.

. Transmission Response from Modeling andeasurements

he transmission of the OADM was measured as aunction of wavelength for the four angles of the tiltedrating, and these results were compared with thoseredicted from modeling. To model the transmis-ion spectrum we used two outputs from theaveguide model: the mode profiles and the effec-

ive indices. We used the transmission dips at theifferent Bragg conditions to estimate the angle thatermits the maximum odd–even reflection whileaintaining a low odd–odd reflection.

. Transmission Spectra

M transmission spectra for the four angles of theilted grating of the OADM were measured with anrbium-doped fiber amplifier as a broadband sourcend an optical spectrum analyzer. As the transmis-ion dips were convolved with the limited bandwidthf the optical spectrum analyzer �0.06 nm�, they hado be measured again at the wavelengths of maxi-um reflection by use of a tunable laser and a pho-

odetector. When power was launched andeasured in the two opposite narrow branches �input

nd output port� of the OADM, two transmission dipsere observed. One dip represents the odd–even

eflection with mode conversion that is necessary torop the channel and the other represents the un-anted odd–odd reflection. The transmission spec-

ra are shown as dashed curves in Figs. 8�a�, 9�a�,

0�a�, and 11�a�. b

170 APPLIED OPTICS � Vol. 43, No. 33 � 20 November 2004

On launching and measuring the power in the twopposite wide branches �drop and add ports� of theADM, we again observed two transmission dips.

n this case they represent the unwanted even–eveneflection and again the desired odd–even reflectionith mode conversion. The transmission curves are

hown as dashed curves in Figs. 8�b�, 9�b�, 10�b�, and1�b�. We obtained cross talk between modes at the-branch transition by launching and measuring theower at opposite branches that have differentidths. The average value of this mode optical cross

alk was approximately 20 dB.The transmission responses from modeling were

btained by use of the overlap integrals and effectivendices that were both computed �complete model�nd measured. In the first case the solid curves inigs. 8�a�, 9�a�, 10�a�, and 11�a� represent the trans-ission response when the power was launched andeasured in the narrow branches. Figures 8�b�,

�b�, 10�b�, and 11�b� show the transmission re-ponses when the power was launched and measured

ig. 8. Transmission at the first angle from complete simulationsolid curves� and from measurements �dashed curves, shifted by40 dB�. Power launched and detected �a� at the narrowranches and �b� at the wider branches.

ig. 9. Transmission at the second angle from complete simula-ion �solid curves� and from measurements �dashed curves, shiftedy 40 dB�. Power launched and detected �a� at the narrow

ranches and �b� at the wider branches.

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n the wider branches. It can be seen from thesegures that the Bragg wavelengths from modelingere slightly shorter in all cases. These differences

�2 nm�, as well as the fact that the modeled moderofiles were wider than the measured profiles, weren indication that the effective indices were higherhan the estimated indices. More-accurate valuesor the effective indices were obtained by use of the

easured Bragg wavelengths and the period of therating in the propagation direction as listed in rowsand 6 of Table 1.The precision of the effective index calculation fromeasurements depends on the precision with which

he optical spectrum analyzer measures wavelengths�0.03 nm� and on the uncertainty in the gratingeriod. The estimated error in the effective indexeasurements was �104: 1.4632 and 1.4637, re-

pectively, for measured and modeled effective indexeff,0 and 1.4586 and 1.4575 for measured and mod-led effective index neff,1.

The average effective index of each mode was com-

ranches and �b� at the wider branches.

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ared with this ion-exchange model. The error inhe modeled effective index was less than 0.05%.

e used the effective indices from the Bragg wave-engths as well as the overlap integral from measure-

ents to obtain the transmission responses of therating as shown by the solid curves in Figs. 12 and3. In the same figures the measured transmissionesponses were plotted again �dashed curves� to makeomparisons easy.

The transmission spectrum at the first angle ��1°�hen the power was launched and measured in theider branches �Figs. 8�b� and 12�b�� shows that the

ven–even reflection �dashed curves at 1.5659 �m� isreater than the odd–even reflection �at 1.5634 �m�.hen the power was launched and measured in the

arrower branches �Figs. 8�a� and 12�a�� the odd–oddeflection �at 1.5607 �m� was also greater than thedd–even reflections. Results from the modelshown as solid curves show good agreement in bothases.

ig. 10. Transmission at the third angle from complete simula-ion �solid curves� and from measurements �dashed curves, shiftedy 40 dB�. Power launched and detected �a� at the narrow

At the second angle ��2°�, when the power was

ig. 11. Transmission at the fourth angle from complete simula-ion �solid curves� and from measurements �dashed curves, shiftedy 40 dB�. Power launched and detected �a� at the narrow

ig. 12. Transmission at the first angle from partial simulationsolid curves� and from measurements �dashed curves, shifted by40 dB�. Power launched and detected �a� at the narrow

ig. 13. Transmission at the second angle from partial simulationsolid curves� and from measurements �dashed curves, shifted by40 dB�. Power launched and detected �a� at the narrow

ranches and �b� at the wider branches.

0 November 2004 � Vol. 43, No. 33 � APPLIED OPTICS 6171

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aunched in the narrow branch a maximum value forhe odd–even reflection was found as expected fromhe model �Figs. 9�a� and 13�a��. In the same figures

slight dip near 1.56 �m could be observed in theeasured transmission owing to cladding modes.t the same angle, the even–even reflection was verytrong, as shown in Figs. 9�b� and 9�b�. At higherngles, results from the model and measurementshowed decreases of reflections strength, as expectedrom their overlap integrals.

Another predictable effect from modeling was thencrease in the wavelengths of maximum reflection�oo, �oe, �ee� when the grating angle was increased.hese increments were found to be proportional to

he changes in the grating period in the propagatingirection ��z � �g�cos ��. In all cases the transmis-ion spectrum showed excellent agreement with theesults of the model. In particular, when the over-ap integrals and effective indices from measure-

ents were used in the model, the differencesetween measurement and model were remarkablymall.

. Transmission Dips As a Function of Grating Angle

ransmission dips were obtained from modeling andeasurements. As we mentioned above, the angles

f the four OADMs relative to one another were ac-urately defined by mask lithography; however, thelignment of the Bragg grating to the sample wouldave provided an error in angle that was constantcross all four OADMs, as the grating was simulta-eously written for all of them. To improve theatch between measured and modeled transmission

ips we reduced the uncertainty that was due to an-ular misalignment ��0.5°� by matching the Braggonditions of different angles and types of reflection:

neff�g � ��1� cos�1 � ε� � ��2� cos�2 � ε�

ig. 14. Transmission dips for top, even–even; middle, odd–even;nd bottom, odd–odd reflection. Solid curves and circles showransmission dips from the complete model and from measure-ents, respectively.

� ��3� cos�3 � ε� � ��4� cos�4 � ε�, (13) p

172 APPLIED OPTICS � Vol. 43, No. 33 � 20 November 2004

here ��1�, ��2�, ��3�, and ��4� are the measured wave-engths in the Bragg condition at the four angles andis the unknown shift that is common to all angles.s the grating period ��g� and the width are the same

or the four waveguides, the effective index was as-umed equal for each. Under this assumption, a nu-erical solution for � � 0.265 was obtained with anncertainty of �0.1°.Figure 14 shows the transmission dips from theodel presented in Sections 2 and 5 �solid curves�

nd from measurements �circles�. From this figuret can be seen that the theoretical transmission dipss a function of angle agree with the measured valueshen the angular correction is made. From model-

ng and measurements, the operation of the OADMan be improved by use of a grating angle of �2.2°.t this angle the output port shows a flat spectrumith only one transmission dip, at �oe, when power is

aunched at the input port �Figs. 9�a� and 13�a��.his indicates that the odd–even reflection necessary

or performing the add–drop function is maximizedt the drop port. The spectrum at this port, never-heless, can also be affected by the power coupledrom the input port as an even mode �Cnee�. A sec-nd reflection peak �Ree� appears in the spectrum, ashown in Figs. 9�b� and 13�b�. This reflection peak isomparable with Roe. However, the effect on theSNR will depend on the magnitude of the even–ven coupling �Cnee�, as shown in Fig. 11. In theabricated samples the cross talk limits the even–ven peak 20 dB below the odd–even peak, as can beeen from measurements and modeling �Fig. 15�.

. Discussion and Summary

he modeled and measured transmission dips showxcellent agreement in magnitude, wavelength, andngle position. The grating angle for maximum re-ection with mode conversion is almost the same ashe angle of minimum odd–odd reflection, as ex-ected from the model.The small errors in the results of the model are due

ig. 15. Spectrum from simulation �solid curves� and measure-ents �solid curves, shifted by 40 dB� at the drop port. The

ower is launched from the input port.

rimarily to the short time of burial: Because this

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ime is short, the effective duration of activity at theppropriate temperature might differ from the onesed in the model. This uncertainty can explain themall discrepancies in the modeled and measuredode profiles and therefore in their overlap integrals.owever, other reasons such as mode cross-couplingoise should also be considered.The relationship between the transmission dips

nd performance parameters such as optical signal-o-noise ratio �OSNR� was described in Refs. 11 and2. A simple estimation of the OSNR from theransmission dip between odd and even modes gave aalue of �25 dB at the output port. Because of modeross talk �Cnee�, the OSNR at the drop port will be20 dB. One can improve the OSNR at output by

ncreasing the strength of the grating while main-aining the bandwidth. To do so requires an in-rease in the grating length and in the value of �n.he OSNR at the drop port, however, can be im-roved by reduction of the mode cross talk at thesymmetric Y branch. It is clear that further im-rovements could be obtained by use of additionallters; simplicity, one of the main advantages of thisype of OADM, would then be lost, however.

Another aspect to consider in this type of opticaldd–drop multiplexer is its polarization dependencePD�. In the measured samples a PD of 0.25 nm wasound. The cause of that PD was attributed to somesymmetry in the index profile of the waveguide.ur measurements show no indication that the re-ection strength is affected by the polarization of theource. Only a shift in the Bragg wavelengths wereound, as expected from our waveguide modeling.onsequently we believe that the PD can be reducedy a deeper burial, postannealing,14 or both.In summary, ion-exchanged asymmetric Y-branchADMs were proposed and demonstrated at accept-ble levels of performance. Their comparative ad-antages lie in their compactness and their relativelyimple fabrication procedure. To improve this typef OADM effectively will require control of the fabri-ation process. Modeling the process from inputanufacturing parameters, such as voltage and tem-

erature, to the final device characteristics, such asransmission response, is a main step in obtaininghe desired control. In this paper we have shownhat it is possible to model the operation of an OADMith an excellent degree of accuracy. Two ways to

btain the transmission characteristics of the deviceere presented. The first procedure predicted the

ransmission dips from the overlap integrals ob-ained from measured odd- and even-mode profiles athe waist. The second procedure produced the sameransmission characteristics, but it used a complete

odel that began with the primary parameters of the

2

on-exchange process. The main sources of errorere described: angular errors during fabrication,oise in the mode profile measurements, and modeross talk at the Y branch. The interrelation of di-erse theories such as ion exchange kinetics, wavequations, and coupled-mode theory have proved toe effective for modeling this device, as can be in-erred from the remarkable agreement of theory witheasurements.

The authors thank the Office of the Vice Presidentor Research and Graduate Studies of the Universityf Arizona for funding in support of this research.

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0 November 2004 � Vol. 43, No. 33 � APPLIED OPTICS 6173